INTERIOR  BALLISTICS 


BY 


JAMES    M.   INGALLS 

Colonel  United  States  Army,  retired 

Formerly  Instructor    of    Ballistics  at    the   U.    S.   Artillery    School  ;    Author    of 

Treatises  on  Exterior  and  Interior  Ballistics,   Ballistic 

Machines,  Ballistic  Tables,  Etc. 


THIRD  EDITION 


NEW  YORK 

JOHN   WILEY  &  SONS 

LONDON:    CHAPMAN  &  HALL,  LIMITED 

1912 


COPYRIGHT  1912 
BY  JAMEvS  M.  INGALLS 


PREFACE  TO  THE  EDITION  OF   1894 

(SECOND  EDITION) 


WHEN,  in  the  summer  of  1889,  it  was  decided  by  the  Staff 
of  the  Artillery  School  to  add  to  the  curriculum  a  course  of 
interior  ballistics,  the  instructor  of  ballistics,  knowing  of  no 
text-book  on  the  subject  in  the  English  language  entirely  suited 
to  the  needs  of  the  school,  employed  the  time  at  his  disposal 
before  the  arrival  of  the  next  class  of  student  officers  in  studying 
up  and  arranging  a  course  of  instruction  upon  this  subject,  so 
important  to  the  artillery  officer.  The  text-book  then  planned 
was  partially  completed  and  printed  on  the  Artillery  School 
press,  and  has  been  tested  by  two  classes  of  student  officers. 

In  the  summer  of  1893  the  author  again  had  leisure  to  work 
on  the  unfinished  text-book,  but  in  the  meantime  he  had  found 
so  much  of  it  which  admitted  of  improvement  that,  with  the 
encouragement  of  Lieutenant-Colonel  Frank,  Second  Artillery, 
the  Commandant  of  the  School,  it  was  decided  to  rewrite  nearly 
the  entire  work  as  well  as  to  complete  it  according  to  the  original 
plan  by  the  addition  of  the  last  two  chapters. 

With  the  exception  of  portions  of  Chapters  IV  and  V,  the 
author  claims  no  originality.  He  has  simply  culled  from  various 
sources  what  seemed  to  him  desirable  in  an  elementary  text- 
book, arranged  it  all  systematically  from  the  same  point  of  view 
and  with  a  uniform  notation. 

ARTILLERY  SCHOOL, 
February  15,  1894. 


258715 


PREFACE  TO  THE  THIRD  EDITION 


THE  second  edition  of  this  work  was  used  as  a  text-book  at 
the  Artillery  School  until  the  School  suspended  operations  at 
the  outbreak  of  war  with  Spain,  in  April,  1898.  This  edition, 
having  become  exhausted,  the  author  has  been  induced  by  the 
request  of  officers  for  whose  wishes  he  has  great  respect,  to 
prepare  a  new  edition  embodying  the  results-of  some  investiga- 
tions which  were  published  in  Volumes  24,  25  and  26  of  the 
Journal  of  the  United  States  Artillery,  and  which  have  been 
favorably  received  by  Artillery  Officers,  both  at  home  and  abroad. 
The  Journal  articles  have  been  rewritten,  and  many  improve- 
ments attempted,  suggested  by  friendly  criticisms  for  which  the 
author  wishes  to  express  his  thanks. 

As  most  of  the  formulas  of  interior  ballistics  in  the  present 
state  of  our  knowledge  of  the  subject  are  more  or  less  empirical 
in  their  nature,  many  applications  of  the  formulas  deduced  in 
Chapter  IV  are  given  in  the  following  chapter  to  show  their 
agreement  with  the  results  of  actual  firing  with  guns  of  widely 
different  calibers.  In  this  connection  it  is  gratifying  to  be  able 
to  quote  from  an  article  published  in  the  Journal  of  the  Royal 
Artillery,  vol.  36,  No.  9,  by  Captain  J.  H.  Hardcastle,  R.A., 
who  states,  with  reference  to  the  formulas  of  Chapter  IV  as 
applied  to  firing-practice  with  English  guns  loaded  with  cordite, 
that  "  After  many  dozens  of  calculations  I  can  find  no  serious 
disagreement  between  the  results  of  calculation  and  experiment." 


VI  PREFACE 

In  this  paper  Captain  Hardcastle  has  very  ingeniously  adapted 
the  formulas  of  Chapter  IV  to  slide- rule  operations,  thereby 
lessening  the  labor  of  calculation  somewhat,  though  at  the 
expense  of  accuracy  in  some  cases.  For  the  benefit  of  those  who 
are  accustomed  to  use  the  slide  rule  in  making  logarithmic  com- 
putations a  supplemental  table  of  the  X  functions  has  been 
added  to  Table  I,  omitting  the  function  X&,  which  is  not  used  in 
Captain  Hardcastle's  method. 

This  work  was  prepared  primarily  for  the  officers  of  our 
Coast  Artillery  Corps;  but  it  is  hoped  that  gun-designers  and 
powder-manufacturers  may  find  in  it  something  useful  to  them. 

The  author  desires  to  express  his  indebtedness  to  Lieut.- 
Colonel  Ormand  M.  Lissak  and  Major  Edward  P.  O'Hern  of 
the  Ordnance  Department  for  valuable  suggestions  and  for 
data  employed  in  the  " applications."  Also  to  Captains  Ennis 
and  Bryant,  for  assistance  in  computing  Table  I. 

PROVIDENCE,  R.  I., 
September  20,  1911. 


TABLE    OF    CONTENTS 


CHAPTER   I 

PAGE 

Definition  and  object. — Early  history  of  gunpowder.  Robins'  experi- 
ments and  deductions.  Mutton's  experiments.  D'Arcy's  method. 
Ram  ford's  experiments  with  fired  gunpowder.  Rodman's  inventions 
and  experiments.  Modern  explosives.  Density  of  powder.  Inflam- 
mation and  combustion  of  a  grain  of  powder.  Inflammation  and 
combustion  of  a  charge  of  powder I  to  14 


CHAPTER   II 

Properties  of  perfect  gases. — Marriotte's  law.  Specific  volume.  Specific 
weight.  Law  of  Gay-Lussac.  Characteristic  equation  of  the  gaseous 
state.  Thermal  units.  Mechanical  equivalent  of  heat.  Specific  heat. 
Specific  heat  of  a  gas  under  constant  pressure.  Specific  heat  under 
constant  volume.  Numerical  value  of  R  for  atmospheric  air.  Law  of 
Dulong  and  Petit.  Determination  of  specific  heats.  Ratio  of  specific 
heats.  Relations  between  heat  and  work  in  the  expansion  of  a  per- 
fect gas.  Isothermal  expansion.  Adiabatic  expansion.  Law  of  tem- 
peratures. Law  of  pressures  and  volumes.  Examples.  Theoretical 
work  of  an  adiabatic  expansion  in  the  bore  of  a  gun.  Noble  and 
Abel's  researches  on  fired  gunpowder  in  close  vessels.  Description 
of  apparatus  employed.  Summary  of  results.  Pressure  in  close  ves- 
sels deduced  from  theoretical  considerations.  Value  of  the  ratio  of 
the  non-gaseous  products  to  the  volume  of  the  charge.  Determina- 
tion of  the  force  of  the  powder,  and  its  interpretation.  Theoretical 
determination  of  the  temperature  of  explosion  of  gunpowder.  Mean 
specific  heat  of  the  products  of  combustion.  Pressure  in  the  bore 
of  a  gun  derived  from  theoretical  considerations.  Table  of  pressures. 
Theoretical  work  effected  by  gunpowder.  Factor  of  effect.  Actual 
work  realized  as  expressed  by  muzzle  energy 1 5  to  54 


CHAPTER   III 

Combustion  of  a  grain  of  powder  under  constant  atmospheric  pressure. — 
Notation.  Definition  of  the  vanishing  surface.  General  expression 
for  the  burning  surface  of  a  grain  of  powder.  Expression  for  the 


VIII  TABLE     OF     CONTENTS 

PAGK 

volume  consumed  in  terms  of  the  thickness  burned.  Definition  of  the 
form  characteristics.  Fraction  of  grain  burned.  Applications. 
Spheres.  Cubes.  Strips.  Solid  cylinders.  Pierced  cylinders. 
Multiperf orated  grains.  General  expression  for  surface  of  combus- 
tion of  multiperforated  grains.  Maximum  surface  of  combustion. 
Slivers.  Expression  for  volume  of  slivers.  Proposed  ratio  of  dimen- 
sions of  multiperforated  grains  to  web  thickness.  Expression  for 
weight  of  charge  burned  at  any  instant.  Expressions  for  initial 
volume  and  surface  of  combustion  of  a  charge  of  powder.  Expression 
for  specific  gravity  of  grain.  Initial  surface  of  unit  weight  of  powder. 
Volume  of  charge.  Gravimetric  density.  Density  of  loading. 
Reduced  length  of  initial  air  space.  Working  formulas  for  English  and 
metric  units.  Examples 55  to  78 


CHAPTER    IV 

Combustion  and  work  of  a  charge  of  powder  in  a  gun. — Introductory 
remarks.  Sarrau's  law  of  burning  under  a  variable  pressure  and 
reason  for  adopting  it.  Expression  connecting  the  velocity  of 
burning  of  grain  with  velocity  of  projectile  in  bore.  Expression  for 
fraction  of  charge  burned  in  terms  of  volumes  of  expansion  of  the 
gases  generated.  Expression  for  velocity  of  projectile  while  powder 
is  burning.  Velocity  of  projectile  after  powder  is  all  burned.  Pres- 
sure on  base  of  projectile  while  powder  is  burning.  Pressure  after 
powder  is  burned.  Expression  for  the  initial  pressure  upon  the  sup- 
position that  the  powder  was  all  burned  before  the  projectile  had 
moved  from  its  seat,  and  the  relation  of  this  pressure  to  the  force 
of  the  powder.  Method  of  computing  the  X  functions.  Special 
formulas.  Expressions  for  maximum  pressure.  Formula  for  velocity 
of  combustion  under  atmospheric  pressure.  Working  formulas.  Eng- 
lish units.  Metric  units.  Characteristics  of  a  powder.  Expressions 
for  constants  in  terms  of  the  characteristics  for  English  and  metric 
units.  Expressions  for  force  of  powder  when  weights  of  charge 
and  projectile  vary 79  to  97 

CHAPTER   V 

Applications. — Formulas  which  apply  only  while  powder  is  burning.  For- 
mulas which  apply  only  after  powder  is  all  burned.  Formulas  which 
apply  at  instant  of  complete  combustion.  Discontinuity  of  pressure 
curve  for  certain  forms  of  grain.  Monomial  formulas  for  velocity 
and  pressure.  Typical  pressure  and  velocity  curves.  Example  of 


TABLE    OF     CONTENTS  ix 

PAGE 

monomial  formulas,  as  applied  to  the  8-inch  B.  L.  R.  Comparison 
of  computed  velocities  and  maximum  pressures  with  observed  values. 
Determination  of  travel  of  projectile  at  point  of  maximum  pressure, 
and  also  when  powder  is  all  burned.  Expression  for  fraction  of 
charge  burned  for  any  travel  of  projectile.  Examples.  Greatest 
efficiency  when  charge  is  all  consumed  at  muzzle.  Application  to 
hypothetical  7-inch  gun.  Binomial  formulas  for  velocity  and  pressure. 
Forms  of  grain  for  which  binomial  formulas  must  be  employed. 
Methods  for  determining  the  constants  from  experimental  firing. 
Applications  to  Sir  Andrew  Noble's  experiments  with  a  6-inch  gun. 
Description  of  the  experiments.  Discussion  of  the  data  for  cordite, 
0.4",  0.35",  and  0.3"  diameter.  Remarks  on  the  so-called  "force  of 
the  powder"  as  deduced  from  the  calculations.  Examples.  Applica- 
tion to  the  Hotchkiss  57-mm.  rapid-firing  gun.  Data  obtained  by 
D'Arcy's  method.  New  method  for  determining  the  form  char- 
acteristics of  the  grains.  Application  to  the  magazine  rifle,  model  of 
1903.  Powder  characteristics.  Formulas  for  designing  guns  for 
cordite,  with  application  to  a  hypothetical  7-inch  gun.  Trinomial 
formulas.  Grains  for  which  these  formulas  are  necessary.  Spherical 
and  cubical  grains.  Formulas  for  computing  the  constants.  Ap- 
plication to  Noble's  experiments  with  ballistite  in  a  6-inch  gun. 
Table  of  computed  velocities  and  pressures.  Remarks  on  the 
velocity  and  pressure  curves.  Examples.  Multiperforated  grains. 
Special  formulas  required  for  these  grains.  Discussion  of  the  data 
obtained  by  the  Ordnance  Board  with  the  6-inch  Brown  wire  gun. 
Remarks  on  the  discontinuity  of  the  pressure  curves.  Examples. 
Superiority  of  uniperforated  to  multiperforated  grains.  Application 
to  the  14-inch  rifle.  Effect  of  increasing  the  volume  of  the  chamber 
upon  the  maximum  pressure.  Better  results  can  be  obtained  by 
lengthening  the  powder  grains.  Table  of  pressures.  .  .  .  9810169 


CHAPTER  VI 

On  the  rifling  of  cannon. — Advantages  of  rifling.  The  developed  groove. 
Uniform  twist.  Increasing  twist.  General  expression  for  pressure 
on  the  lands.  Angular  acceleration.  Pressure  for  uniform  twist. 
Increasing  twist.  Semi-cubical  parabola.  Common  parabola.  Rel- 
ative width  of  grooves  and  lands.  Application  to  the  lo-inch 
B.  L.  R.,  model  of  1888.  Application  to  the  14-inch  gun.  Retarding 
effect  of  a  uniform  twist  of  one  turn  in  twenty-five  calibers.  .  170  to  186 

Tables  18910215 


INTERIOR  BALLISTICS 


CHAPTER  I 
INTRODUCTION 

Definition  and  Object. — Interior  ballistics  treats  of  ihe\ 
formation,  temperature  and  volume  of  the  gases  into  which  the 
powder  charge,  in  the  chamber  of  a  gun,  is  converted  by  com- 
bustion, and  the  work  performed  by  the  expansion  of  these 
gases  upon  the  gun,  carriage  and  projectile.  Its  object  is  the 
deduction  and  discussion  of  rules  and  formulas  for  calculating 
the  velocity,  both  of  translation  and  of  rotation,  which  the 
gases  of  a  given  weight  of  powder  of  known  composition  and 
quality  are  able  to  impart  to  a  projectile  and  their  reaction 
upon  the  gun  and  carriage.  The  discussion  of  the  formulas 
deduced  will  bring  out  many  important  questions,  such  as  the 
proper  relation  of  weight  of  charge  to  weight  of  projectile  and 
length  of  bore,  the  best  size  and  shape  of  the  powder  grains  for 
different  guns  and  their  effect  upon  the  maximum  and  muzzle 
pressures,  the  velocity  of  recoil,  etc.  The  most  approved 
formulas  for  calculating  the  pressures  upon  the  surf  ace  »of  the 
bore  will  be  given ;  but  the  methods  which  have  been  devised 
for  building  up  the  gun,  so  as  best  to  resist  these  pressures,  will 
not  be  entered  upon  here  as  their  consideration  belongs  to 
another  branch  of  the  subject. 

Early  History  of  Interior  Ballistics. — For  more  than  five 
hundred  years  gunpowder — an  intimate  mixture  of  nitre, 
sulphur  and  charcoal, — was  used  almost  exclusively  as  the  pro- 


2  INTERIOR   BALLISTICS 

pelling  agent  in  firearms;  and  though  it  has  been  entirely 
superseded  within  the  last  quarter  of  a  century  by  gun-cotton, 
mtro-glycerine,  and  their  various  compounds,  yet  it  possessed 
many  admirable  qualities  which  the  modern  powders  do  not 
as  yet  so  fully  enjoy.  It  ignited  easily  without  deflagration; 
its  effects  were  regular  and  sure;  its  manufacture  was  economical, 
rapid  and  comparatively  safe;  it  produced  but  little  erosion 
in  the  bore.  Finally,  it  kept  well  in  transportation,  and  in- 
definitely in  properly  ventilated  magazines.  It  is  on  record 
that  experiments  made  with  gunpowder,  manufactured  more 
than  two  centuries  before,  showed  that  it  had  lost  none  of  its 
ballistic  qualities.  The  principal  objection  to  gunpowder,  as 
compared  to  nitrocellulose  powders,  are  the  dense  volumes  of 
smoke  accompanying  its  explosion,  the  fouling  of  the  bore, 
and  the  comparatively  large  charges  required  to  give  the 
desired  muzzle  velocity,  necessitating  an  abnormal  enlarge- 
ment of  the  powder  chamber  or  an  impracticable  lengthening 
of  the  gun. 

Robins'  Experiments  and  Deductions. — The  celebrated  Ben- 
jamin Robins  seems  to  have  been  the  first  investigator  who  had  a 
tolerably  correct  idea  of  the  circumstances  relating  to  the  action 
and  force  of  fired  gunpowder.  In  a  paper  which  was  read  before 
the  Royal  Society  in  1743  entitled,  "New  principles  of  gunnery," 
Robins  described  among  other  things  some  experiments  he  had 
made  for  determining  the  velocities  of  musket  balls  when  fired 
with  given  charges  of  powder.  These  velocities  were  measured 
by  means  of  the  ballistic  pendulum  invented  by  Robins,  "the 
idea  of  which  is  simply  that  the  ball  is  discharged  into  a  very 
large  but  movable  block  of  wood,  whose  small  velocity,  in  conse- 
quence of  that  blow,  can  be  easily  observed  and  accurately 
measured.  Then,  from  this  small  velocity  thus  obtained,  the 
large  one  of  the  ball  is  immediately  derived  from  this  simple 
proportion,  viz.,  as  the  weight  of  the  ball  is  to  the  sum  of  the 
weights  of  the  ball  and  the  block,  so  is  the  observed  velocity  of 


INTRODUCTION  3 

the  last  to  a  fourth  proportional,  which  is  the  velocity  of  the  ball 
sought."  * 

The  deductions  which  Robins  makes  from  these  experiments, 
so  far  as  they  relate  to  interior  ballistics,  may  be  summarized 
as  follows: 

(1)  Gunpowder  fired  either  in  a  vacuum  or  in  air  produces, 
by  its  combustion,  a  permanent  elastic  fluid  or  air. 

(2)  The  pressure   exerted  by   this  fluid  is,  cateris  paribus, 
directly  as  its  density. 

(3)  The  elasticity  of  the  fluid  is  increased  by  the  heat  it 
has  at  the  time  of  explosion. 

(4)  The  temperature  of  the  fluid  at  the  moment  of  combus- 
tion is  at  least  equal  to  that  of  red-hot  iron. 

(5)  The  maximum  pressure  exerted  by  the  fluid  is  equal  to 
about  1,000  atmospheres. 

(6)  The  weight  of  the  permanent  elastic  fluid  disengaged  by 
the  combustion  is  about  three-tenths  that  of  the  powder,  and  its 
volume  at  ordinary  atmospheric  temperature  and  pressure  is 
about  240  times  that  occupied  by  the  charge. 

These  deductions,  considering  the  extremely  erroneous  ancj 
often  absurd  opinions  that  were  entertained  by  those  who 
thought  upon  the  subject  at  all  in  Robins'  time — and  even  down 
to  the  close  of  the  century — show  that  Robins  is  well  entitled 
to  be  called  the  "father  of  modern  gunnery." 

Button's  Experiments. — Dr.  Charles  Hutton,  professor  of 
mathematics  in  the  Royal  Military  Academy,  Woolwich,  con- 
tinued Robins'  experiments  at  intervals  from  1773  to  1791.  He 
improved  and  greatly  enlarged  the  ballistic  pendulum  so  that 
it  could  receive  the  impact  of  i -pound  balls,  whereas  that  used 
by  Robins  was  adapted  for  musket  balls  only.  Button's 
experiments  are  given  in  detail  in  his  thirty-fourth,  thirty-fifth, 
thirty-sixth,  and  thirty-seventh  tracts.  They  verify  most  of 

*  Hutton's  "  Mathematical  Tracts,"  vol.  3,  p.  210  (Tract  37),  London,  1812. 


4  INTERIOR   BALLISTICS 

Robins'  deductions,  but  with  regard  to  Robins'  estimate  of  the 
temperature  of  combustion  and  the  maximum  pressure  Hutton 
says:  "This  was  merely  guessing  at  the  degree  of  heat  in  the 
inflamed  fluid,  and,  consequently,  of  its  first  strength,  both 
which  in  fact  are  found  to  be  much  greater."  *  His  own  estimate 
of  the  temperature  is  double  that  of  Robins,  and  he  places  the 
maximum  pressure  of  fired  gunpowder  at  2,000  atmospheres. 
Hutton  gives  a  formula  for  the  velocity  of  a  spherical  projectile 
at  any  point  of  the  bore,  upon  the  assumption  that  the  combus- 
tion of  the  charge  is  instantaneous  and  that  the  expansion  of 
the  gas  follows  Mariotte's  law — no  account  being  taken  of  the 
loss  of  heat  due  to  work  performed — a  principle  which  at  that 
time  was  unknown. 

D'Arcy's  Method. — In  1760  the  chevalier  D'Arcy  sought  to 
determine  the  law  of  pressure  of  the  gas  in  the  bore  of  a  musket 
by  measuring  the  velocity  of  the  projectile  at  different  points 
of  the  bore.  This  he  accomplished  by  successively  shortening 
the  length  of  the  barrel  and  measuring  for  each  length  the  velocity 
of  the  bullet  by  means  of  a  ballistic  pendulum.  Having  obtained 
from  these  experiments  the  velocities  of  the  bullets  for  several 
different  lengths  of  travel,  the  corresponding  accelerations  could 
be  calculated,  and  then  the  pressures,  by  multiplying  the 
accelerations  by  the  mass.  This  was  the  first  attempt  to 
determine  the  law  of  pressures  dynamically. 

Rumford' s  Experiments  with  Fired  Gunpowder. — The  first 
attempt  to  measure  directly  the  pressure  of  fired  gunpowder 
was  made,  in  1792,  by  our  countryman,  the  celebrated  Count 
Rumford.  A  most  interesting  account  of  his  experiments  is 
given  in  his  memoir  entitled  "Experiments  to  determine  the 
force  of  fired  gunpowder,"  f  which  must  be  regarded  as  the  most 
important  contribution  to  interior  ballistics  wrhich  had  been 

*  Tracts,  vol.  3,  p.  211. 

t  Philosophical  Transactions,  London,  1797,  p*.  222;  also  "The  Complete 
Works  of  Count  Rumford,"  Boston,  1870,  vol.  I,  p.  98. 


INTRODUCTION  5 

made  up  to  that  time.  The  apparatus  used  by  Rumford  con- 
sisted of  a  small  and  very  strong  wrought-iron  mortar  (or 
eprouvette),  which  rested  with  its  axis  vertical  upon  a  solid 
stone  foundation.  This  mortar  (or  barrel,  as  Rumford  calls  it), 
was  2.78  inches  long  and  2.82  inches  in  diameter  at  its  lower 
extremity  and  tapered  slightly  toward  the  muzzle.  The  bore 
(or  chamber)  was  cylindrical,  one-fourth  of  an  inch  in  diameter 
and  2.13  inches  deep.  At  the  centre  of  the  bottom  of  the  barrel 
there  was  a  projection  0.45  inch  in  diameter  and  1.3  inches  long, 
having  an  axial  bore  0.07  inch  in  diameter  connecting  with  the 
chamber  above,  but  closed  below,  forming  a  sort  of  vent,  but 
having  no  opening  outside. 

By  this  arrangement  the  charge  could  be  fired  without  any 
loss  of  gas  through  the  vent  by  the  application  of  a  red-hot 
ball  provided  with  a  hole,  into  which  the  projecting  vent-tube 
could  be  inserted,  which  latter  would  thus  become  in  a  short 
time  sufficiently  heated  to  ignite  the  powder.  The  upper  part 
of  the  bore  or  muzzle  was  closed  by  a  stopper  made  of  compact, 
well-greased  sole  leather,  which  was  forced  into  the  bore,  until 
its  upper  surface  was  flush  with  the  face  of  the  mortar,  and  upon 
this  was  placed  the  plane  surface  of  a  solid  hemisphere  of  hard- 
ened steel,  whose  diameter  was  1.16  inches.  "Upon  this 
hemisphere  the  weight  made  use  of  for  confining  the  elastic 
fluid  generated  from  the  powder  in  its  combustion  reposed. 
This  weight  in  all  the  experiments,  except  those  which  were 
made  with  very  small  charges  of  powder,  was  a  piece  of  ordnance 
of  greater  or  less  dimensions  or  greater  or  less  weight,  according 
to  the  force  of  the  charge,  placed  vertically  upon  its  cascabel 
upon  the  steel  hemisphere  which  closed  the  end  of  the  barrel; 
and  the  same  piece  of  ordnance,  by  'having  its  bore  filled  by  a 
greater  or  smaller  number  of  bullets,  as  the  occasion  required, 
was  made  to  serve  for  several  experiments." 

*  Rumford's  Works,  vol.  i,  p.  121. 


6  INTERIOR   BALLISTICS 

As  one  of  the  objects  of  Rumford's  experiments  was  to 
determine  the  relation  between  the  pressure  of  the  powder  gases 
and  their  density,  he  varied  the  charge,  beginning  with  i  grain, 
and  for  each  charge  placed  a  weight,  which  he  judged  was  about 
equivalent  to  the  resulting  pressure,  upon  the  hemisphere.  If, 
on  firing,  the  weight  was  lifted  sufficiently  to  allow  the  gases  to 
escape,  it  was  increased  for  another  equal  charge;  and  this  was 
repeated  until  a  weight  was  found  just  sufficient  to  retain  the 
gaseous  products  —  that  is,  so  that  the  leathern  stopper  would 
not  be  thrown  out  of  the  bore,  but  only  slightly  lifted.  The 
density  of  the  powder  gases  could  easily  be  determined  by 
comparing  the  weight  of  the  charge  with  the  weight  of  powder 
required  to  completely  fill  the  chamber  and  vent,  which  latter 
was  about  25^2  grains  troy.  Rumford  increased  the  charges  a 
grain  at  a  time  from  i  grain  to  18  grains,  and  from  a  mean  of 
all  the  observed  pressures  he  deduced  the  empirical  formula, 


i  +.0004* 


in  which  p  is  the  pressure  in  atmospheres  and  x  the  density  of 
loading  to  a  scale  of  1000  —  that  is,  for  a  full  chamber  x  =  1000; 
for  one-half  full  x  =  500,  and  so  on.  This  formula  gives 
29,178  atmospheres  for  the  maximum  pressure  —  that  is,  when 
the  powder  entirely  fills  the  space  in  which  it  is  fired.  In  this 
case  the  value  of  x  is  1000,  and  Rumford's  pressure  formula 
becomes 

p  =  1.841  X  1000  I<4  =  29178 

Nearly  a  century  later  Noble  and  Abel  (see  Chapter  II) 
found  by  their  experiments,  which  are  entirely  similar  in  charac- 
ter to  those  of  Rumford,  that  the  maximum  pressure  of  fired 
gunpowder  is  but  6,554  atmospheres,  or  43  tons  per  square  inch; 
and  this  result  has  been  accepted  by  all  writers  on  interior 


INTRODUCTION  7 

ballistics  as  being  very  near  the  truth.     Their  formula  for  the 
pressure  in  terms  of  Rumford's  x  is 

2.818* 
^  '"  1  —  0.00057* 

in  which  p  and  x  are  denned  as  before.     If  in  this  formula  we 
make  x  =  1000,  we  have,  as  already  stated, 

2.818  X  1000 

P  =  —  ""  =  6554 

i  —  0.57 

For  small  densities  of  loading,  Noble  and  Abel's  formula 
gives  greater  pressures  than  Rumford's  principally  because  the 
powder  used  by  the  later  investigators  was  the  stronger;  but 
as  the  densities  increase  this  is  reversed.  With  a  charge  of  18 
grains,  for  which  x  =  702,  Noble  and  Abel's  formula  gives  a 
pressure  of  3,298  atmospheres,  while  Rumford's  gives  8,140 
atmospheres.  To  enable  us  to  understand  the  cause  of  this 
great  difference  in  the  results  obtained  by  these  eminent  savants 
(which  is  very  instructive),  we  will  go  a  little  into  detail.  Two 
experiments  were  made  by  Rumford  with  a  charge  of  18  grains 
of  powder.  In  the  first  of  these  a  24-pounder  gun,  weighing 
8,08 1  pounds,  was  placed  vertically  on  its  cascabel  upon  the 
steel  hemisphere  closing  the  muzzle  of  the  barrel.  When  ihe 
charge  was  fired  "the  weight  was  raised  with  a  very  sharp 
report,  louder  than  that  of  a  well-loaded  musket."  The  barrel 
was  again  loaded  with  18  grains  as  before,  and  enough  shot  were 
placed  in  the  bore  of  the  24-pounder  gun  to  increase  its  weight 
to  8,700  pounds.  Upon  firing  the  powder  by  applying  the 
red-hot  ball  "the  vent-tube  of  the  barrel  was  burst,  the  explosion 
being  attended  with  a  very  loud  report."  These  experiments 
were  the  eighty-fourth  and  eighty-fifth,  and  closed  the  series. 

In  the  eighty-fourth  experiment  a  weight  of  8,081  pounds 
was  actually  raised  by  the  explosion  of  18  grains  of  powder 
(about  one-fourth  the  service  charge  of  the  Springfield  rifle), 


8  INTERIOR   BALLISTICS 

acting  upon  a  circular  area  one-quarter  of  an  inch  in  diameter. 
To  raise  this  weight  under  the  circumstances  would  require  a 
pressure  of  more  than  11,200  atmospheres,  while,  as  we  have 
seen,  the  actual  pressure  due  to  this  density  of  loading,  according 
to  Noble  and  Abel's  formula,  is  but  3,298  atmospheres.  Evident- 
ly then  the  weight  in  this  experiment  was  not  raised  by  mere 
pressure;  but  we  must  attribute  a  great  part  of  the  observed 
effect  (in  consequence  of  the  position  of  the  charge  at  the  bottom 
of  the  bore)  to  the  energy  with  which  the  products  of  combustion 
impinged  against  the  leathern  stopper,  which  had  only  to  be 
raised  0.13  inch  (the  thickness  of  the  leather)  to  allow  the  gases 
to  escape.  In  Noble  and  Abel's  experiments  there  was  no  such 
blow  from  the  products  of  combustion  because  the  apparatus 
for  determining  the  pressure  (crusher  gauge)  was  placed  within 
the  charge.  Had  the  leathern  stopper  in  Rumford's  experiments 
been  a  little  longer,  it  is  probable  that  his  conclusions  would 
have  been  entirely  different. 

Rodman's  Inventions  and  Experiments. — We  have  space 
only  to  mention  the  names  of  Gay-Lussac,  Chevreul,  Graham, 
Piobert,  Cavalli,  Mayevski,  Otto,  Neumann,  and  others,  who 
did  original  work,  of  more  or  less  value,  for  the  science  of  interior 
ballistics  prior  to  the  year  1860.  We  will,  however,  dwell  a 
few  moments  on  the  important  work  done  by  Captain  (after- 
wards General)  T.  J.  Rodman,  of  our  own  Ordnance  Department, 
between  the  years  1857  and  1861.*  The  objects  of  Rodman's 
experiments  were:  First,  to  ascertain  the  pressure  exerted 
upon  different  points  of  the  bore  of  a  4 2 -pounder  gun  in  firing 
under  various  circumstances.  Second,  to  determine  the  press- 
ures in  the  y-inch,  Q-inch,  and  n-inch  guns  when  the  charges 
of  powder  and  the  weight  of  projectiles  were  so  proportioned 
that  there  should  be  the  same  weight  of  powder  behind,  and 


*  "  Experiments  on  Metal  and  Cannon  and  Qualities  of  Cannon  Powder," 
by  Captain  T.  J.  Rodman,  Boston,  1861. 


INTRODUCTION  9 

the  same  weight  of  metal  in  front  of  each  square  inch  of  area 
of  cross-section  of  the  bore.  Third,  to  determine  the  differences 
in  pressure  and  muzzle  velocity  due  to  the  variations  in  the 
size  of  the  powder  grains;  and,  fourth,  to  determine  the  pressures 
exerted  by  gunpowder  burned  in  a  close  vessel  for  different 
densities  of  loading. 

For  the  purpose  of  carrying  out  these  experiments  Rodman, 
instead  of  using  the  system  of  varying  weights  employed  by 
Rumford,  invented  what  he  called  the  "indenting  apparatus/' 
which  has  since  been  extensively  used,  not  only  in  this  country 
but  in  all  foreign  countries,  under  the  name  of  Rodman's  pressure 
(or  cutter)  gauge;  and  which  is  too  well  known  to  require  a 
description. 

The  maximum  pressure  of  gunpowder  when  exploded  in  its 
own  space,  as  determined  by  Rodman  by  the  bursting  of  shells 
filled  with  powder,  ranged  from  4,900  to  12,600  atmospheres; 
the  mean  of  all  the  experiments  giving  8,070  atmospheres,  or 
53  tons  per  square  inch.  These  results,  though  much  nearer  the 
truth  than  those  deduced  by  Rumford,  are  still  about  25  per 
cent,  greater  than  Noble  and  Abel's  deductions;  and  this  is 
undoubtedly  due  to  the  position  of  the  pressure  gauge,  which 
was  placed  near  the  exterior  surface  of  the  shell,  so  that  when 
the  products  of  combustion  had  reached  the  gauge  they  had 
acquired  a  considerable  energy  which  probably  exaggerated  the 
real  pressure.  The  same  causes,  it  will  be  remembered,  vitiated 
Rumford's  experiments.  In  both  cases  it  was  as  if  a  charge  of 
small  shot  had  been  fired  with  great  velocity  against  the  leathern 
stopper  in  the  one  case,  or  the  end  of  the  piston  of  the  indenting 
tool  in  the  other. 

General  Rodman  was  the  first  person  to  suggest  the  proper 
shape  for  powder  grains,  in  order  to  diminish  the  initial  velocity 
of  emission  of  gas  and  to  more  nearly  equalize  the  pressure  in 
the  bore  of  the  gun/  For  this  purpose  he  employed  what  he 
termed  a  "perforated  cake  cartridge"  composed  of  disks  of 


10  INTERIOR   BALLISTICS 

compressed  powder  from  i  to  2  inches  thick  and  of  a  diameter 
to  fit  the  bore.  Rodman  demonstrated  that  such  a  form  of 
cartridge  would  relieve  the  initial  strain  by  exposing  a  minimum 
surface  at  the  beginning  of  combustion,  while  a  greater  volume 
of  gas  would  be  evolved  from  the  increasing  surfaces  of  the 
cylindrical  perforations  as  the  space  behind  the  projectile  be- 
came greater;  and  this  would  tend  to  distribute  the  pressure 
more  uniformly  along  the  bore.  Rodman's  experiments  with 
this  powder  in  the  15 -inch  cast-iron  gun  which  he  had  recently 
constructed  for  the  government — and  which  is  without  doubt  the 
most  effective  and  the  best  smooth-bore  gun  ever  made — fully 
confirmed  his  theory;  but  for  many  reasons  he  found  it  more 
convenient  and  equally  satisfactory  to  build  up  the  charge  by 
layers  of  pierced  hexagonal  prisms  about  an  inch  in  diameter 
fitting  closely  to  one  another,  instead  of  having  them  of  the 
diameter  of  the  bore. 

The  war  of  the  rebellion  which  was  inaugurated  while 
General  Rodman  was  in  the  midst  of  his  discoveries  and  in- 
ventions, put  an  end  forever  to  his  investigations,  but  his  ideas 
were  speedily  adopted  in  Europe,  and  his  " prismatic  powders," 
but  slightly  modified,  are  extensively  used. 

Modern  Explosives. — Gun-cotton,  made  by  immersing 
cleaned  and  dried  cotton  waste  in  a  mixture  of  strong  nitric  and 
sulphuric  acids,  was  discovered  by  Schonbein  of  Basel,  in  1846, 
who  immediately  proposed  to  employ  it  as  a  substitute  for 
gunpowder.  General  von  Lenk  made  many  experiments  with 
gun-cotton  by  compressing  it  into  cubes  or  cylinders,  with  the 
idea  of  employing  it  for  artillery  use.  But  all  his  efforts  failed 
from  the  fact  that,  no  matter  how  much  it  was  compressed,  it 
was  still  mechanically  porous;  and  when  ignited  in  a  gun  the 
flame  and  hot  gases  speedily  penetrated  the  mass  causing  it  to 
detonate,  or,  at  least,  to  approach  dangerously  near  to  detona- 
tion. It  was  not  until  the  discovery  in  the  early  eighties  that 
gun-cotton  could  be  dissolved  or  made  into  a  paste,  or  colloid, 


INTRODUCTION  1 1 

by  acetone  and  other  so-called  solvents,  that  it  was  possible  to 
employ  it  as  a  propellant.  In  this  condition,  when  moulded 
into  grains  and  thoroughly  dried,  it  loses  its  mechanical  porosity 
and  burns  from  the  surface  in  parallel  layers,  the  grain  retaining 
its  original  form  until  completely  consumed. 

Gun-cotton  is  mixed  in  certain  proportions  with  nitro- 
glycerine to  form  nearly  all  the  powders  employed  for  war 
purposes.  For  example,  the  powder  used  in  the  British  army 
and  navy  (called  cordite),  consists  of  65  per  cent,  of  gun-cot- 
ton, 30  per  cent,  of  nitro-glycerine  and  5  per  cent,  of  mineral 
jelly  or  vaseline,  this  latter  being  used  as  a  preservative.  This 
is  also  very  nearly  the  composition  of  the  powder  used  in  the 
United  States  army  and  navy.  For  a  full  account  of  the 
properties,  manufacture  and  uses  of  gun-cotton  and  nitro- 
glycerine, the  reader  is  referred  to  General  Weaver's  "Notes 
on  Explosives." 

Density  of  Powder. — By  density  of  a  powder  is  meant  its 
specific  gravity,  or  the  ratio  of  the  weight  of  a  given  volume  of 
the  powder  to  the  weight  of  an  equal  volume  of  water  at  the 
standard  temperature.  It  is  sometimes  referred  to  as  mercurial 
density,  since  it  may  be  determined  by  art  apparatus  which 
utilizes  the  property  of  mercury  of  filling  the  interstices  between 
the  grains  without  penetrating  into  the  pores  or  uniting  chemi- 
cally with  the  powder.  The  density  varies  somewhat  according 
to  the  pressure  to  which  the  grains  were  subjected  during  the 
manufacture  and  ranges  from  about  1.56  to  1.65. 

Inflammation  and  Combustion  of  a  Grain  of  Powder. — 
Inflammation  is  the  spreading  of  the  flame  over  the  free  surface 
of  the  grain  from  the  point  of  ignition.  Combustion  is  the 
propagation  of  the  burning  into  the  interior  of  the  grain.  Igni- 
tion is  produced  by  the  sudden  elevation  of  the  temperature 
of  a  small  portion  of  the  grain  to  about  180°  C.  (in  the  case  of 
cordite)  either  by  contact  with  an  ignited  body,  by  mechanical 
shock  or  friction,  or  by  detonation  of  a  fulminate.  The  velocity 


12  INTERIOR   BALLISTICS 

of  inflammation  depends  upon  the  nature  of  the  source  of  heat 
which  ignites  it,  upon  the  state  of  the  surface  of  the  grain  and 
upon  its  density  and  dryness.  The  combustion  of  a  grain  takes 
place  in  successive  concentric  layers,  and  in  free  air  equal 
thicknesses  are  burned  in  equal  times.  As  the  mass  of  gas 
disengaged  in  any  given  time  is  proportional  to  the  quantity  of 
powder  burned  during  the  same  time,  and,  therefore,  propor- 
tional to  the  surface  of  inflammation,  it  follows  that  the  emission 
of  gas  is  largely  influenced  by  the  form  of  the  grain.  For 
example,  if  the  grain  is  spherical  the  surface  of  inflammation 
decreases  rapidly  up  to  the  end  of  its  burning  where  it  is  zero. 
On  the  other  hand,  the  surface  of  inflammation  (or  of  combus- 
tion) of  a  multi-perforated  grain  increases  until  it  is  nearly 
consumed. 

Inflammation  and  Combustion  of  a  Charge  of  Powder.— 
The  inflammation  of  a  charge  of  powder  involves  the  trans- 
mission of  the  flame  from  one  grain  to  another.  Its  velocity 
depends  not  only  upon  the  inflammability  of  the  grain  but  also 
upon  the  facility  with  which  the  gases  first  generated  are  able  to 
penetrate  the  charge.  This  is  assisted  by  a  proper  arrangement 
of  the  grains  composing  the  charge  and  also  by  placing  an  igniter 
of  fine  rifle  powder  at  each  end  of  the  cartridge.  The  com- 
bustion of  a  charge  composed  of  grains  of  the  same  form  and 
dimensions  should,  from  what  has  been  said,  practically  termi- 
nate at  the  same  time  with  each  or  any  grain;  and,  therefore, 
the  time  of  combustion  of  a  charge  increases  with  the  size  of  the 
grains,  and  is  in  all  cases  with  service  powders  much  longer  than 
the  time  of  inflammation. 

If  a  charge  of  powder  be  confined  in  a  close  vessel  and  ignited, 
its  combustion  takes  place  silently,  and  permanent  gases  and  a 
certain  amount  of  solid  matter  are  produced  which  can  be 
collected  for  analysis  by  opening  the  vessel,  as  in  the  experiments 
of  Noble  and  Abel  described  in  Chapter  II.  In  this  case  no 
work  is  performed  by  the  gases,  and  the  accompanying  phe- 


INTRODUCTION  13 

nomena  are  comparatively  simple.  But  if  the  combustion  takes 
place  in  a  chamber  of  which  one  of  the  walls  is  capable  of  moving 
under  the  tension  of  the  gases,  which  condition  is  realized  in 
cannon,  the  resulting  phenomena  are  much  more  complicated, 
as  a  little  consideration  will  show. 

When  the  charge  of  powder  in  the  chamber  of  a  loaded  gun 
is  ignited  at  both  ends  of  the  cartridge,  all  the  grains  will  be 
inflamed  practically  simultaneously.  The  first  gaseous  products 
formed  will  expand  into  the  air-spaces  of  the  chamber  and 
almost  immediately  acquire  a  tension  sufficient  to  start  and 
overcome  the  forcing  of  the  projectile.  This  latter  once  in 
motion  will  encounter  no  resistances  in  the  bore  comparable  with 
those  which  opposed  its  start,  and  its  velocity  will  rapidly  in- 
crease under  the  continued  action  of  the  pressure  of  the  gases. 
This  pressure  will  also  increase  at  first;  for,  though  the  displace- 
ment of  the  projectile  gives  a  greater  space  for  the  expanding 
gases,  this  is  more  than  compensated  for  by  a  more  abundant 
disengagement  of  gas.  But  the  pressure  soon  reaches  its 
maximum;  for  if,  on  the  one  hand,  the  disengagement  of  gas 
is  accelerated  by  the  increase  of  pressure,  on  the  other  hand  the 
increasing  velocity  of  the  projectile  offers  more  and  more  space 
for  the  gases  to  expand  in.  The  velocity  itself  would  soon 
reach  a  maximum  if  the  bore  were  sufficiently  prolonged;  for 
in  addition  to  the  friction  and  the  resistance  of  the  air,  both  of 
which  retard  the  motion  of  the  projectile,  the  propulsive  force 
decreases  by  the  expansion  and  cooling  of  the  gases.  Therefore 
the  retarding  forces  will  in  time  predominate  and  the  projectile 
be  brought  to  rest.  Its  velocity  starting  from  zero  passes  to  its 
maximum  and  if  the  gun  terminated  at  this  point  the  projectile 
would  leave  the  bore  with  the  greatest  velocity  the  charge  was 
capable  of  communicating  to  it. 

So  far  only  charges  in  general  have  been  considered.  Take, 
now,  a  charge  composed  of  small  grains  of  slight  density.  The 
initial  surface  of  inflammation  will  be  very  great  and  the  emission 


14  INTERIOR   BALLISTICS 

of  gas  correspondingly  abundant.  The  pressure  will  increase 
rapidly,  and,  in  consequence,  the  velocity  of  combustion.  It 
results  from  this  that  the  grains  will  be  consumed  nearly  as  soon 
as  inflamed,  and  this  before  the  projectile  has  had  time  to  be 
displaced  by  a  sensible  amount.  Hence  all  the  gases  of  the 
charge,  disengaged  almost  instantaneously,  will  be  confined  an 
instant  within  the  chamber;  their  tension  will  be  very  great, 
and  they  will  exert  upon  the  walls  of  the  gun  a  sudden  and  violent 
force  which  may  rupture  the  metal,  and  which  in  all  cases  will 
produce  upon  the  gun  and  carriage  shocks  which  are  destructive 
to  the  system  and  prejudicial  to  accuracy  of  fire.  On  the  other 
hand  the  projectile  will  be  thrown  quickly  forward,  as  by  a 
blow  from  a  hammer. 

If,  on  the  contrary,  the  charge  is  made  up  of  large  grains  of 
great  density,  the  total  weight  of  gas  emitted  will  be  the  same  as 
before;  but  the  mode  of  emission  will  be  different.  The  initial 
surface  of  inflammation  will  be  less,  and  the  initial  tension  of  the 
gas  not  so  great.  The  combustion  will  take  place  more  slowly, 
and  will  be  only  partially  completed  when  the  projectile  shall 
have  begun  to  move.  The  pressure  of  the  gases  will  attain  a 
maximum  less  than  in  the  preceding  case,  but  the  pressure  will 
decrease  more  slowly.  Under  the  continued  action  of  this 
pressure,  the  velocity  of  the  projectile  will  be  rapidly  accelerated 
and  at  the  muzzle  will  differ  but  little  from  that  obtained  by  the 
fine  powder,  without  producing  upon  the  gun  and  carriage  the 
destructive  effects  mentioned  above. 


CHAPTER  II 

PROPERTIES  OF  PERFECT  GASES 

Mariotte's  Law.  —  When  a  mass  of  gas  is  subjected  to  pressure 
the  volume  diminishes  until  the  increased  tension  just  balances 
the  pressure;  and  it  was  found  by  experiment  that  if  the  tem- 
perature of  the  gas  remains  constant,  the  tension,  or  pressure, 
is  inversely  proportional  to  the  volume.  Thus,  if  Vi  and  v2 
represent  different  volumes  of  the  same  mass  of  gas  and  pi  and 
p2  the  corresponding  tensions,  or  pressures,  then  if  the  tem- 
perature is  the  same  for  both  volumes  we  have  the  proportion: 


Hence 

Vi  pi  =  v-2  p-2  =  constant. 

That  is,  for  every  mass  of  gas  at  invariable  temperature  the 
product  of  the  volume  and  tension  is  constant.  This  law  is 
generally  called  Mariotte's  law,  though  it  was  first  discovered 
by  the  English  chemist  Robert  Boyle,  in  1662,  and  verified  by 
Mario  tte  in  1679. 

Specific  Volume.  —  The  specific  volume  of  a  gas  is  the  volume 
of  unit  weight  at  zero  temperature  and  under  the  normal  atmos- 
pheric pressure.  Designate  the  specific  volume  by  v0  and  the 
normal  atmospheric  pressure  by  p0.  Then  we  have  by  Mariotte's 
law 


Specific  Weight.  —  The  specific  weight  of  a  gas  is  the  weight 
of  unit  volume  at  zero  temperature  and  under  the  pressure  p0. 
It  is  therefore  the  reciprocal  of  the  specific  volume  v0. 

15 


1 6  INTERIOR   BALLISTICS 

Law  of  Gay-Lussac. — The  coefficient  of  expansion  of  a  gas 
is  the  same  for  all  gases,  and  is  sensibly  constant  for  all  tem- 
peratures and  pressures.  Let,  as  before,  v0  be  the  specific 
volume,  vt  the  volume  at  temperature  /  and  a  the  coefficient 
of  expansion.  Then  the  variation  of  volume  by  Gay-Lussac's 
law  will  be  expressed  by  the  equation 

Vt  -  vo  =  atv0; 
whence 

vt  =  V0  (i  +  at) 

The  value  of   the    coefficient  a   is   approximately  -     -    for 

273 

each  degree  centigrade.     The  last  equation  may,  therefore,  be 
written 

'  i  H ^ 


Characteristic  Equation  of  the  Gaseous  State. — The  last 
equation,  which  expresses  Gay-Lussac's  law,  may  be  combined 
with  Mariotte's  law,  introducing  the  pressure  p.  The  problem 
may  be  enunciated  as  follows :  Having  given  the  specific  volume 
of  a  gas  v0  to  determine  its  volume  vt  at  a  temperature  t  under 
the  corresponding  pressure  pt. 

Let  x  be  the  volume  vt  would  become  at  o°  C.,  under  the 
pressure  pt.  Then  by  Gay-Lussac's  law 

vt  =  x  (  i  +  a  t) 
and  by  Mariotte's  law 

Pt*  =  POVO', 
whence  eliminating  x, 

Pt  »/=  PoVo  (i  +  «  0    =  (273  +  0 


c.  oo. 

Since  -    -  is  constant,  put 

273. 


7?     **. 

K  =  ~ 

273 


PROPERTIES  OF  PERFECT  GASES  17 

whence 

ptvt  =  £(273  +  /); 

or,  dropping  the  subscripts  as  no  longer  necessary, 

pv  =  R(2J3  +  t) 


The  temperature  (273  +  /)  is  called  the  absolute  tempera- 
ture, and  is  reckoned  from  a  zero  placed  273  degrees  below  the 
zero  of  the  centigrade  scale.  Calling  the  absolute  temperature 
T  there  results  finally 

pv  =  RT         ......      (  i) 

which  is  called  the  characteristic  equation  of  the  gaseous  state. 
It  is  simply  another  expression  of  Mariotte's  law  in  which  the 
temperature  of  the  gas  is  introduced. 

Equation  (i)  expresses  the  relation  existing  between  the 
pressure,  volume  and  absolute  temperature  of  a  unit  weight  of 
gas.  For  any  number  y  units  of  weight  occupying  the  same 
volume  v  the  relation  evidently  becomes 

pv=yRT      ......      (2) 

A  gas  supposed  to  obey  exactly  the  law  expressed  in  equation 
(i)  is  called  a  perfect  gas,  or  is  said  to  be  theoretically  in  the 
perfectly  gaseous  state.  This  condition  represents  an  ideal 
toward  which  gases  approach  more  nearly  as  their  state  of 
rarefaction  increases.  Of  all  gases,  hydrogen  approximates 
most  closely  to  such  an  hypothetical  substance,  though  at 
ordinary  temperatures  the  simple  gases,  nitrogen,  oxygen  and 
atmospheric  air,  may  for  most  practical  purposes  be  considered 
perfect  gases. 

Thermal  Unit.  —  The  heat  required  to  raise  the  temperature 
of  unit  weight  of  water  at  the  freezing  point  one  degree  of  the 
thermometer  is  called  a  thermal  unit.  There  are  two  thermal 
units  in  general  use,  namely:  the  British  thermal  unit  (B.  T.  U.), 
which  is  the  heat  required  to  raise  the  temperature  of  one  pound 


1 8  INTERIOR   BALLISTICS 

of  water  from  32°  F.  to  33°  F.;  and  the  French  thermal  unit 
(called  a  calorie),  which  is  the  heat  required  to  raise  the  tem- 
perature of  one  kilogram  of  water  from  o°  C.  to  i°  C.  There 
is  still  another  thermal  unit  of  frequent  use,  namely:  the  heat 
required  to  raise  the  temperature  of  one  pound  of  water  from 
o°  C.  to  i°  C.,  and  which  may  be  designated  as  the  pound- 
centigrade  (P.  C.)  unit. 

Mechanical  Equivalent  of  Heat. — The  mechanical  equivalent 
of  heat  is  the  work  equivalent  of  a  thermal  unit,  and  will  be 
designated  by  E.  According  to  Rowland  the  value  of  E  is 
778  foot-pounds  for  a  B.  T.  U.  Since  a  degree  of  the  centi- 
grade scale  is  —  of  a  degree  of  the  Fahrenheit  scale,  we  have 

for  a  P.  C.  thermal  unit,  E  =  —  X  778  =  1400.4  foot-pounds. 

3 

Also  since  there  are  3.280869  feet  in  a  metre,  the  value  of  E  for 
a  calorie  is 

1400  4 
3. 280869  =  42<5'84  kil°gram-metres- 

Specific  Heat. — The  quantity  of  heat,  expressed  in  thermal 
units,  which  must  be  imparted  to  a  unit  weight  of  any  sub- 
stance to  increase  its  temperature  one  degree  of  the  thermometer, 
or  the  quantity  of  heat  given  up  by  the  substance  while  its 
temperature  is  lowered  one  degree,  is  called  its  specific  heat. 
The  specific  heat  of  different  substances  varies  greatly.  Thus, 
if  a  pound  of  mercury  and  a  pound  of  water  receive  the  same 
quantity  of  heat  the  temperature  of  the  former  will  be  much 
greater  that  the  latter.  Indeed,  it  requires  about  32  times  as 
much  heat  to  raise  the  temperature  of  water  i°  as  it  does  to 
raise  the  temperature  of  mercury  by  the  same  amount. 

The  heat  imparted  to  a  substance  is  expended  in  three 
different  ways:  i.  Increasing  the  temperature,  which  may 
be  called  vibration  work;  2.  In  doing  internal  or  disgregation 
work;  3.  In  doing  external  work  by  expansion.  If  it  were 


PROPERTIES  OF  PERFECT  GASES  IQ 

possible  to  eliminate  the  two  latter,  we  should  get  the  true 
specific  heat,  or  the  heat  necessary  to  increase  the  temperature 
simply.  For  a  perfect  gas,  however,  the  disgregation  work  is 
zero,  and  for  all  substances  the  disgregation  work  is  small  in 
comparison  with  the  vibration  work.  The  specific  heat  of  a 
gas  may  be  determined  in  two  different  ways,  giving  results 
which  are  of  fundamental  importance  in  thermodynamics, 
namely:  Specific  heat  under  constant  pressure,  and  specific 
heat  under  constant  volume. 

Specific  Heat  of  a  Gas  Under  Constant  Pressure. — To 
fix  the  ideas  suppose  a  unit  weight  of  gas  to  be  confined  in  a 
spherical  envelope  capable  of  expanding  without  the  expenditure 
of  work  and  which  allows  no  heat  the  gas  may  have  to  escape, 
and  to  be  in"  equilibrium  with  the  constant  pressure  of  the  at- 
mosphere. Under  these  conditions  let  a  certain  quantity  of 
heat  be  applied  to  the  gas  just  sufficient  to  raise  its  temperature 
one  degree  of  the  thermometer  after  it  has  expanded  until 
equilibrium  is  again  restored.  This  quantity  of  heat,  in  thermal 
units  (designated  by  Cp),  is  called  specific  heat  under  constant 
pressure. 

Specific  Heat  Under  Constant  Volume. — Next  repeat  the 
experiment  just  described,  but  replacing  the  elastic  envelope, 
which  by  hypothesis  permitted  the  gas  to  expand  freely,  by  a 
rigid  envelope,  thus  keeping  the  volume  of  the  gas  constant 
while  heat  is  applied.  It  will  now  be  found  that  there  will  less 
heat  be  required  to  raise  the  temperature  of  the  gas  one  degree. 
The  quantity  of  heat  required  in  this  case  is  called  the  specific 
heat  under  constant  volume,  and  in  terms  of  the  thermal  unit 
employed,  is  designated  by  Cv. 

The  number  of  molecules  of  gas  being  the  same  in  both 
experiments  and  the  temperatures  being  equal,  it  follows  that 
the  quantity  of  heat  absorbed  by  the  gas,  or  the  vibration  work, 
is  the  same  in  both  experiments.  But  in  the  experiment  made 
under  constant  volume  the  heat  absorbed  is  necessarily  equal 


20  INTERIOR  BALLISTICS 

to  the  total  heat  supplied,  namely,  Cv  thermal  units,  since  the 
envelope  is  considered  impermeable  to  heat.  Therefore  in  the 
first  experiment  there  is  a  loss  of  heat  equal  to  Cp  —  Cv  thermal 
units.  This  last  heat  must  then  have  been  expended  in  over- 
coming the  atmospheric  pressure  in  expanding;  and  the  work 
done  will  be  found  by  multiplying  Cp  —  Cv  by  the  mechanical 
equivalent  of  heat.  That  is,  for  an  increase  of  one  degree  of 
temperature, 

Work  of  expansion  =  (Cp  -  Cv)  E. 

The  work  of  overcoming  a  constant  resistance  is  measured 
by  the  product  of  the  resistance  into  the  path  described.  In 
the  case  of  the  expanding  gas  just  considered  the  constant  re- 
sistance is  the  atmospheric  resistance  p0\  and  the  path  described 
is  measured  by  the  increase  of  volume  of  the  gas.  To  determine 
this  latter  Gay-Lussac's  law  gives  for  the  centigrade  scale 

tV0 

Vt   —  VQ    =    '  - 
273 

and  therefore  for  an  increase  of  temperature  of  one  degree  there 
is  an  increase  of  volume  equal  to  ^0/273.  The  work  of  expansion 
for  one  degree  is,  therefore, 

* 


273 

The  quantity  R  is,  then,  the  external  work  of  expansion 
performed  under  atmospheric  pressure  by  unit  weight  of  gas 
when  its  temperature  is  raised  one  degree  centigrade.  But 
this  work  of  expansion  has  already  been  found  equal  to  (Cp  —  Cv) 
E.  There  results,  therefore,  the  important  equation 

(CV-CV}E  =  ^  =  R      .     .     .     .     (3) 

for  the  centigrade  scale  of  temperature.     For  the  Fahrenheit 
scale  the  equation  becomes 

(Cp  -CV)E  =  ^-—  fa1) 

491.4 


PROPERTIES  OF  PERFECT  GASES  21 

Numerical  Value  of  R. — The  numerical  value  of  R  for  any 
particular  gas  depends  upon  the  units  of  length  and  weight 
adopted,  the  atmospheric  pressure,  the  specific  weight  of  the 
gas  and  the  scale  of  temperature.  Throughout  this  chapter 
the  foot  and  pound  will  be  employed  for  the  units  of  length  and 
weight,  respectively;  and  generally  the  centigrade  scale  of 
temperature  will  be  used.  The  adopted  value  of  the  atmos- 
pheric pressure  is 

Po  =  10333  kgs.  per  m.2  log  =  4-01423- 
p0  =  2116.3  Ibs.  per  ft.2  log  =  3-32558- 
p0  =  14.6967  Ibs.  per  in.3  log  =  1.16722. 

As  an  example,  find  the  numerical  value  of  R  for  atmospheric 
air.  The  specific  weight  of  this  gas,  according  to  the  best 
authorities,  is  0.080704  Ibs.  The  specific  volume  is  the  recip- 
rocal of  this;  or  V0  =  12.3909  c.  ft.  Therefore, 

2116.3  X  12.3000 
R  = —       JV     =  96.056  foot-pounds. 

273 

Therefore,  for  one  pound  of  this  gas, 

p  v  =  96.056  T] 
and  for  y  pounds 

p  v  =  96.056  y  T. 

Law  of  Dulong  and  Petit. — The  product  of  the  specific  heat 
of  a  perfect  gas  under  constant  volume,  by  its  density,  is  a  constant 
number. 

By  the  density  of  a  gas  is  meant  its  specific  weight  expressed 
in  terms  of  the  specific  weight  of  atmospheric  air  taken  as  unity. 
If  Cva  is  the  specific  heat  of  air  at  constant  volume  and  Cv  and  d 
the  specific  heat  at  constant  volume,  and  density,  respectively, 
of  any  other  gas,  then  in  accordance  with  this  law, 

C,d  =  Cm. 

Determination  of  Specific  Heats. — The  specific  volume 
and  the  specific  heat  at  constant  pressure  of  a  gas  can  both  be 


22 


INTERIOR   BALLISTICS 


determined  with  great  accuracy  by  experiment;  but  the  specific 
heat  under  constant  volume  is  almost  impossible  to  measure 
directly  on  account  of  the  dissipation  of  heat  through  the  sides 
of  the  vessel  containing  the  gas.  It  can,  however,  be  computed 
by  equation  (3)  which  gives 

Cv  =  Cp  -  -§ (4) 

By  means  of  this  equation  and  the  direct  determination  of 
specific  heats  under  constant  pressure,  Regnault  has  deduced 
the  following  law  for  perfect  gases : 

The  specific  heats  under  constant  pressure  and  constant  volume 
are  independent  of  the  pressure  and  volume. 

The  following  table  gives  the  specific  weights,  volumes  and 
heats  of  those  gases  which  approximate  most  nearly  to  the 
theoretically  perfect  gas.  The  values  of  R  were  computed 
by  (i)  and  those  of  Cv  by  (4).  The  temperature  is  supposed 
to  be  o°  C.,  and  the  barometer  to  stand  at  760  mm.  =  29.922  in. : 


Gas 

Specific 
Weight 

Specific 
Volume 

R 

CP 

Cv 

Atmospheric  air 

Pounds 
0.080704 

Cubic  Feet 
12.3909 

96  .  056 

0.23751 

o.  16892 

Nitrogen  

0.078394 

12.7569 

98.887 

0.24380 

O.I73I9 

Oxygen  
Hydrogen 

0.089230 

o  OOSSQO 

II  .2070 
178  8910 

86.878 
1386.8 

0.21751 
3  .  40900 

0.15547 
2.41873 

Ratio  of  Specific  Heats. — In  the  study  of  interior  ballistics 
the  values  of  Cp  and  Cv  for  the  gases  given  off  by  the  explosion 
of  the  charge  are  of  little  importance.  It  suffices  generally  to 
know  their  ratio  which  is  constant  for  perfect  gases  and  approxi- 
mately so  for  all  gases  at  the  high  temperature  of  explosion. 
That  this  ratio  is  constant  for  perfect  gases  may  be  shown  as 
follows :  Since 


PROPERTIES  OF  PERFECT  GASES  23 

R  =  P°V°  =   P°   =    P° 

273       273  w0      273  d  wa 

in  which  wa  is  the  specific  weight  of  atmospheric  air,  we  shall 
have  for  two  gases  distinguished  by  accents,  the  relation 


that  is,  the  values  of  R  for  two  perfect  gases  are  inversely  as 
their  densities.  But  by  the  law  of  Dulong  and  Petit  we  have 

C,     d"      R' 

T^r  =  ~rf  —  ~n,  (as  shown  above). 

Therefore 

R'       R" 

-~r  =  r^r  =  constant. 

^    V  ^     V 

Therefore  from  equation  (4), 

c*  /? 

-:£-  =  i  +  -T^-E  =  constant  =  n  (say). 

Cv  C^zi 

If  we  compute  n  by  means  of  atmospheric  air,  we  shall  have 

96.056 

n  =  i  +  —  -77)  -  —       —  =  1.406. 
0.16892  X  1400.4 

Relations  Between  Heat  and  Work  in  the  Expansion  of 
Perfect  Gases.  —  The  relations  which  exist  between  the  varia- 
tions of  the  volume  and  pressure  of  a  given  weight  of  gas  and 
the  heat  necessary  to  produce  them,  may  now  be  determined 
from  equation  (i)  as  follows:  This  equation  is 

pv  =  RT 

and  contains  three  arbitrary  variables  p,  v  and  T.  If  we  suppose 
an  element  of  heat,  d  q,  to  be  applied  to  the  gas,  the  temperature 
will  generally  be  augmented  by  an  elementary  amount  d  T, 
and  this  may  be  accomplished  in  three  different  ways  : 

i.  The  volume  may  increase  by  the  element  dv  without 
altering  the  pressure.  2.  The  pressure  may  increase  by  d  p 


24  INTERIOR    BALLISTICS 

while  the  volume  remains  constant.  3.  The  volume  and 
pressure  may  both  vary  at  the  same  time.  We  will  consider 
each  of  these  cases  separately. 

i.  Differentiating  (i),  supposing  p  constant,  we  have 


and  therefore  the  quantity  of  heat  communicated  to  the  gas 
will  be,  in  thermal  units,  from  the  definition  of  specific  heat, 

,/         r  IT      CpPdv 
dq  =  CpdT         —   — 


2.  If,    the   volume   v   remaining   constant,    the   pressure    is 
varied  by  d  p,  we  shall  have,  proceeding  as  before, 

A         r  ir       CvvdP 
a  q  =  Lv  a  1  —  5  — 

3.  If  the  volume    and  pressure    vary  together,   the  corre- 
sponding element  of  heat  will  be  the  sum  of  the  partial  variations 
given  above.     That  is 

dq  =  -j(Cppdv  +  Cvvdp)       ...     (5) 

The  differential  of  (i)  is 

RdT  =  pdv  +  vdp;     .      .      .      .      (6) 
whence,  eliminating  v  d  p  between  (5)  and  (6),  there  results 

dq  =  CvdT  +  Cp~RCv  pdv        ...      (7) 
Whence,  since  Cp,  Cv  and  R  are  constants  for  the  same  gas, 

/c  —  c  r 
dT+-^—  J  pdv. 

The  first  integral  represents  the  change  of  temperature  and 
the  second  the  external  work  of  expansion.  Denoting  by  7\ 
and  T  the  initial  and  final  temperatures  of  the  expanding  gas 
and  by  W  the  external  work,  we  have 

q  =  (7\  -  T)  Cv  +  -^  W     .      .      .      (8) 


PROPERTIES  OF  PERFECT  GASES  25 

Isothermal  Expansion. — If  we  suppose  the  initial  temperature 
TI  to  remain  constant,  that  is,  that  just  sufficient  heat  is  im- 
parted to  the  gas  while  it  expands  to  maintain  its  initial  tem- 
perature, equation  (8)  becomes 


We  see  in  this  case  that  the  quantity  of  heat  absorbed  by 
the  gas  is  proportional  to  the  external  work  done.  The  quantity 

r> 

-~r -— ^r  is,  therefore,  the  ratio  of  the  effective  work  of  a  unit 

weight  of  gas  to  the  quantity  of  heat  absorbed,  or  the  mechanical 
equivalent  of  heat,  E.     Therefore 

E  =  c^c, 

a  result  already  established  by  another  method. 

The  work  performed,  therefore,  by  the  isothermal  expansion 
of  unit  weight  of  gas  is  given  by  the  equation 

W  =  E  q  =  1400.4  q  foot-pounds,      ...      (9) 

where  q  is  expressed  in  P.  C.  thermal  units. 

The  work  of  an  isothermal  expansion  may  also  be  expressed 
in  terms  of  the  initial  and  final  volumes  or  pressures.  Thus, 
substituting  in  the  general  equation  of  the  work  of  expansion, 

W  =--  fpdv, 

the  value  of  p  from  (i)  and  integrating  between  the  limits  Vi 
and  v,  we  have 

W  =  R  7\  log,  ^  =  #1  »i  log,  ~    •     v     •     (10) 

where  v  is  the  greater  volume  and  Vi  the  less. 
Since  from  (i) 

^__!i 

Vi    '   p 


26  INTERIOR   BALLISTICS 

we  also  have 

W  =  plvlloge^     .....     (n) 

in  which  pi  is  the  greater  tension  and  p  the  less. 

The  reciprocal  of  E  may  be  called  the  heat  equivalent  of 
work,  that  is,  the  quantity  of  heat  equivalent  to  a  unit  of  work. 
Therefore  from  (9),  (10)  and  (n),  we  have 

W        Pl  Vl   W    v   ' 
q~-=~E    :     ~E~  log^| 

p^  Pl  '      '      •      '     (»> 

£     loge    p          j 

Equations  (10)  and  (n),  by  inverting  the  ratios  of  volumes 
or  pressures,  evidently  hold  good  when  the  initial  volume  Vi 
and  initial  tension  pi  are  changed  by  compression  under  constant 
temperature  into  the  less  volume  v  and  greater  tension  p. 

Adiabatic  Expansion.  —  If  a  gas  expands  and  performs  work 
in  an  envelope  impermeable  to  heat,  so  that  it  neither  receives 
nor  gives  up  heat  during  the  expansion,  the  transformation  is 
said  to  be  adiabatic.  In  such  an  expansion  the  temperature 
and  tension  of  the  gas  both  diminish  and  the  work  performed 
must  be  less  than  for  an  isothermal  expansion,  other  things 
being  equal.  For  an  adiabatic  expansion,  q  is  zero  in  (8)  and, 
therefore,  since  the  temperature  diminishes, 


PF  =  7r(rI-r)i 

l^p      I-'? 

=  _*_  .     .... 

=  CVE  (iV-  T)\ 

Therefore  in  an  adiabatic  expansion  the  work  done  is  pro- 
portional to  the  fall  of  temperature. 

Next  consider  equation  (7),  where,  if  we  make  dq  zero,  it 
becomes 


PROPERTIES  OF  PERFECT  GASES  27 

/?  T 

which,  by  dividing  by  Cv  and  replacing  p  by  its  value ,     re- 
duces to 

_  dT  dv 

~T  ~  ~       ~v 

Integrating  between  limits,  we  have 
T 
T,  " 

Again,  making  d  q  zero  in  (5),  we  have 

o  =  Cp  p  d  v  +  Cvv  d  p, 
which  may  be  written  (dividing  by  Cv  p  v) 

d  v       dp 

Integrating  between  limits,  we  have 


Combining  (14)  and  (15)  gives  the  important  relations 

-  •  •  •  <•«> 


By  means  of  (16)  the  work  of  an  adiabatic  expansion  given 
by  (13)  may  be  expressed  either  in  terms  of  the  initial  and 
terminal  volumes,  or  of  the  initial  and  terminal  pressures.  Thus, 
since 


the  last  of  equations  (13)  may  be  written, 


28  INTERIOR  BALLISTICS 

EXAMPLES. 

1 .  Determine  the  volume  of  5  pounds  of  oxygen  at  a  pressure 
of  50  pounds  per  square  inch  by  the  gauge,  and  at  a  temperature 
of  60°  C. 

The  real  pressure  is  gauge  pressure  plus  the  atmospheric 
pressure  =  50  +  14.6967  =  64.6967  Ibs.  per  in.2  Therefore, 
p  =  144  X  64.6967  Ibs.  per  ft.2  T  =  273  +  60  =  333°.  R  = 
86.878.  Therefore  from  (2), 

5  X  86.878  X  333  ,,3 

144  X  64.6967 

2.  One  pound  of  atmospheric  air  occupying  a  volume  of 
one  cubic  foot  has  a  tension  of  50,000  Ibs.  per  ft.2     What  is  its 
temperature  by  the  Fahrenheit  scale  ? 

For  the  centigrade  scale  we  have  R  =  96.056,  v  =  i,  p  = 
50,000  and  y  =  i. 


Therefore  T  =       ~  =  52O.°54  C.  -  968.097  F. 
.*.  /   =  968.97  -  491.4  =  477-°57  F. 

3.  A  gas-receiver  having  a  volume  of  3  cubic  feet  contains 
half  a  pound  of  oxygen  at  70°  F.     What  is  the  pressure  by  the 
gauge  ? 

Here  y  =  ^,  v  =  3,  R  =  86.876,  t  =  21°  -  C.,  and  T  =  294!. 

Therefore, 

86.876  X  2941- 
^  =   %  v  7  v  TA,~  ~  J4-697  =  14-876  Ibs.  per  in. 

£   A   3   A   144 

4.  A  spherical  balloon  20  feet  in  diameter  is  to  be  inflated 
with  hydrogen  at  60°  F.,  when  the  barometer  stands  at  30.2  in., 
so  that  gas  may  not  be  lost  on  account  of  expansion  when  the 


PROPERTIES  OF  PERFECT  GASES  29 

balloon  has  risen  till  the  barometer  stands  at  19.6  in.,  and  the 
temperature  falls  to  40°  F.  How  many  pounds  and  how  many 
cubic  feet  of  gas  are  to  be  run  in? 

Here  v  =  --  *  X  io3  =  4188.8  ft.3 

=  *9-6  X  *"6-3  =  I386.8  lbs.  per  ft.' 

29.9215 

T  =  277!  C. 
R  =  1386.8. 

p  V 
•'•  y  =  ~D~T  =  I5-°92  lbs. 


To  determine  the  number  of  cubic  feet  of  gas  run.  in,  we  have 

yR  T 
v  =  S-j—  =  2827.4  ft.3, 

where 

30.  2  X  2116.3 


29.9215 


=  2136.0. 


2P=  -1  (60  _  32)  +  273  =  288f. 

5.  "The  balloon  in  which  Wellman  intends  to  seek  the  North 
Pole  has  a  capacity  of  2  24,244  cubic  feet,  and  weighs,  with  its 
car  and  machinery,  6,600  lbs.     What  will  be  its  lifting  capacity 
when  filled  with  hydrogen  at  10°  C.  and   760  mm.  of  the  ba- 
rometer ?"     (Lissak's  "  Ordnance  and  Gunnery,"  p.  61.) 

The  balloon,  when  inflated,  will  hold  at  10°  C.,  17,458  lbs. 
of  air  and  1,209  Mbs.  °f  hydrogen.  Its  lifting  capacity  will. 
therefore,  be  17,458  —  (1,209  +  6,600)  =  9,649  lbs. 

6.  Two  pounds  of  air  expand  adiabatically  from  an  initial 
temperature  of  60°  F.,  and  a  pressure  of  65.3  lbs.  per  in.2  to  a 
pressure  of  50  lbs.  per  in.2     Determine  the  initial  and  terminal 
volumes,    the   terminal   temperature    and    the    external    work 
done. 


30  INTERIOR   BALLISTICS 

Here    p,  =  144  X  65.3  =  9403-2;    p  =  50  X  144  ==  7200; 
T!  =  288!  C.;  R  =  96.056;  y  =  2.     Take  n  =  1.4 

.'.0i=—  T— -  =  5-8954  ft.3 
ri 


T  =  T\~)  7  =  267.37  C.  =  481.266  F. 

:.t  =  21.87  F. 
W  =  ^    -  (Ti-  T)  =  10177  ft.-lbs. 

7.  Compute  the  work  of  expansion  of  2  pounds  of  air  at 
temperature    100°    C.,    which    expands    adiabatically   until   it 
doubles   its   volume.     Also   determine   the   temperature   after 
expansion  and  the  ratio  of  the  initial  and  terminal  pressures. 

Answers:  W  =  43378  ft.-lbs. 

/  =  i9°.68  C. 
P  =  0.3789 />L 

8.  A  mass  of  air  occupying  a  volume  of  3  ft.3  expands  adiabati- 
cally from  an  initial  temperature  of  70°  F.,  and  pressure  of 
85  Ibs.  per  in.2,  until  external  work  of  8,000  ft.-lbs.  has  been 
done.     Compute  the  terminal  volume,  pressure,  temperature, 
and  weight  of  air. 

Answers:  v  =  3.768  ft.3 

p  =  61.78  Ibs.  per  in.2 
t  =  23°.86  F. 
y  =  i. 3  Ibs. 

Theoretical  Work  of  an  Adiabatic  Expansion  in  the  Bore  of 
a  Gun. — If,  in  the  first  of  equations  (17),  we  replace  Cv  E  7\,  by 

its  equal  —     -  it  becomes  for  y  pounds  of  gas 


n—i 


n—  i 


(  v 


PROPERTIES  OF  PERFECT  GASES  3! 

This  equation  gives  the  work  of  y  pounds  of  gas  at  the  initial 
temperature  7\,  expanding  from  the  initial  volume  v±  to  volume 
v.  Suppose  the  mass  of  gas  to  occupy  the  chamber  of  a  gun 
with  the  projectile  at  its  firing  seat;  and  to  expand  by  forcing 
the  projectile  along  the  bore.  In  this  case  v\  will  be  the  volume 
of  the  chamber,  which  is  an  enlargement  of  the  bore,  and  is 
measured  by  what  is  called  the  reduced  length  of  the  chamber; 
that  is,  by  the  length  of  a  cylinder  whose  cross-section  is  the 
same  as  the  bore  and  whose  volume  is  that  of  the  chamber. 
If  u0  is  the  reduced  length  required,  Vc  the  volume  of  the  chamber 
and  d  the  diameter  corresponding  to  the  area  of  cross-section 
of  bore,  and  which  on  account  of  the  rifling  is  slightly  greater 
than  the  caliber,  we  evidently  have 


~d2 

The  variable  volume  v  is  the  volume  of  the  chamber  plus 
the  volume  of  the  bore  in  rear  of  the  projectile  after  it  has 
moved  any  distance  u;  and  is,  therefore,  measured  by  u0  -f  u. 
Therefore  the  above  expression  for  the  work  of  expansion  be- 
comes 

W  =  y—^\i  - 
n-i    ( 

There  is  some  uncertainty  as  to  the  proper  value  of  n  for 
the  gases  of  fired  powder.  As  we  have  seen,  the  value  of  this 
ratio  for  perfect  gases  is  approximately  1.4;  and  it  has  been 
generally  assumed  that  at  the  high  temperature  of  combustion 
of  powder  the  gases  formed  may  be  regarded  as  possessing  all 
the  properties  of  perfect  gases ;  and  therefore  most  of  the  earlier 
writers  on  interior  ballistics  employed  this  value  of  n  in  their 
deductions.  But  more  recent  experiments  have  shown  that 
this  value  is  too  great,  but  have  not  fixed  its  true  value.  The 
experiments  of  Noble  and  Abel  with  the  gases  of  fired  gunpowder, 
at  or  near  the  temperature  of  combustion,  made  n  =  i^  nearly; 


32  INTERIOR   BALLISTICS 

and  this  is  the  value  which,  for  want  of  a  better,  we  will  adopt 
in  what  follows.  Introducing  this  value  of  n  into  the  above 
expression  for  the  work  of  expansion;  and  making  R  Tl  =  f 
and  the  ratio  u/u0=  x,  we  have 


•s/*|'--TrnFl 

The  work  of  expansion  in  the  bore  of  a  gun  is  expended  in 
many  ways,  but  chiefly  in  the  energy  of  translation  imparted 
to  the  projectile.  If  we  assume  that  the  entire  work  is  thus 
expended,  we  shall  have 


W  W       (  (l   +JC3    ) 

It  is  evident  from  (2)  that  /  is  the  pressure  per  unit  of 
surface  of  unit  weight  of  gas  at  temperature  7\.  The  ratio  x 
is  the  number  of  volumes  of  expansion  of  y  pounds  of  gas  due 
to  the  travel  u. 

The  assumption  that  the  work  of  expansion  is  measured  by 
the  energy  of  translation  of  the  projectile  does  not  change  the 
form  of  the  second  member  of  (19);  and  it  is  evident  that  by 
giving  to  /  a  suitable  value  determined  by  experiment,  the 
equality  expressed  in  (19)  may  be  strictly  true.  But  in  this 
case  /  ceases  to  have  the  value  R  7\  and  becomes  simply  an 
experimental  coefficient. 

In  English  units  (pound  and  foot)  ,  /  would  be  theoretically 
the  pressure  in  pounds  per  square  foot  of  one  pound  of  gas  at 
temperature  T\  confined  in  a  volume  of  one  cubic  foot. 

In  metric  units  (kilogramme  and  decimetre),  /  would  be 
defined  as  above,  making  the  proper  change  of  units. 

We  may  deduce  a  second  approximation  to  the  velocity 
impressed  upon  the  projectile  by  the  expansion  of  the  gas  by 
taking  into  account  the  work  performed  upon  the  gun  and 
carriage,  as  well  as  upon  the  projectile.  We  will  suppose  the 
gun  mounted  upon  a  free-recoil  carriage.  Let  M  be  the  mass 


PROPERTIES  OF  PERFECT  GASES  33 

of  the  gun  and  carnage,  V  their  velocity  at  any  period  of  mo- 
tion and  m  the  mass  of  the  projectile.  The  expression  for  the 
work  of  expansion  will  now  be 

2  W  =  m  IT  +  M  V\    .      .      .      .      .     (20) 

A  second  equation  between  the  velocities  v  and  V  can  be 
deduced  by  equating  the  momenta  of  the  system  proiected  upon 
the  axis  of  the  gun.  We  thus  obtain 

m  v  =  M  V      ......     (21) 

Eliminating  V  from  (20)  and  (21)  there  results 

2W 


This  expression  for  v2  is  the  same  as  that  given  by  (19)  with 
the  exception  of  the  small  fraction  m/M  which  can  be  safely 
neglected  in  comparison  with  unity.  Similarly  it  may  be  shown 
that  the  work  expended  upon  the  projectile  in  giving  it  rota- 
tion about  its  axis  is  small  in  comparison  with  the  work  of 
translation. 

Noble  and  Abel's  Researches  on  Fired  Gunpowder.  —  Noble 
and  Abel's  experiments  on  the  explosion  of  gunpowder  in  close 
vessels  were  given  to  the  world  in  two  memoirs  which  were 
read  before  the  Royal  Society  in  1874  and  1879,  respectively. 
These  experiments  have  an  important  bearing  upon  the  subject 
of  interior  ballistics,  since  they  furnish  the  most  reliable  values 
we  possess  of  the  temperature  of  combustion  of  fired  gun- 
powder, the  mean  specific  heat  of  the  products  of  combustion 
(solid  as  well  as  gaseous),  the  ratio  of  solid  to  gaseous  products, 
and,  lastly,  what  is  known  as  the  force  of  the  powder,  —  all  of 
which  are  important  factors  in  computing  the  work  done  by  the 
gases  of  a  charge  of  gunpowder  exploded  in  the  chamber  of  a  gun. 

The  vessels  in  which  the  explosions  were  produced  were  of 
two  sizes,  the  smaller  one  for  moderate  charges  and  for  experi- 


34 


INTERIOR   BALLISTICS 


ments  connected  with  the  measurement  or  analysis  of  the  gases, 
while  in  the  larger  one  Captain  Noble  states  that  he  has  succeeded 
in  absolutely  retaining  the  products  of  combustion  of  a  charge  of 
23  pounds  of  gunpowder.*  These  vessels  consisted  of  a  steel 
barrel  open  at  both  ends,  the  two  open  ends  being  closed  by 
carefully  fitted  screw  plugs  (firing  plug  and  crusher  plug), 
furnished  with  gas  checks  to  prevent  any  escape  of  gas  past  the 
screw.  In  the  firing  plug  was  a  conical  hole  closed  from  within 
by  a  steel  cone  which  was  ground  into  its  place  with  great 
exactness,  and  which,  when  the  cylinder  was  prepared  for  firing, 
was  covered  with  very  fine  tissue  paper  to  give  it  electrical  insula- 
tion from  the  rest  of  the  apparatus.  The  two  wires  from  a 
Leclanche  battery  were  attached,  the  one  to  the  insulated  cone 
and  the  other  to  the  firing  plug,  and  were  connected  within  the 
powder  chamber  by  a  fine  platinum  wire  passing  through  a 
glass  tube  filled  with  mealed  powder.  This  platinum  wire 
became  heated  when  the  electric  current  passed  through  it,  and 
the  charge  was  thus  fired.  At  the  opposite  end  of  the  cylinder 
from  the  firing  plug  was  another  plug  fitted  with  a  crusher 
gauge  for  determining  the  pressure  of  the  gases.  The  vessel 
was  also  provided  with  an  arrangement  for  collecting  the  gases 
after  an  explosion  for  analysis,  measurement  of  quantity,  or 
for  other  purposes. 

Results    of   the   Experiments. — It   was   found    that   about 

I  57  per  cent,  by  weight  of  the  products  of  combustion  were  non- 
gaseous,  consisting  principally  of  potassium  carbonate,  potassium 

j  sulphate,  and  potassium  sulphide,  the  first  named  greatly 
preponderating.  The  remaining  43  per  cent,  were  permanent 
gases,  principally  C02,  CO  and  N.  These  gases,  when  brought 
to  a  temperature  of  o°  C.,  and  under  the  normal  atmospheric 
pressure  of  760  millimetres,  occupied  about  280  times  the  volume 
of  the  unexploded  powder. 


*  Lecture  on  Internal  Ballistics,  by  Captain  Noble,  London,  1892,  p.  12. 


PROPERTIES  OF  PERFECT  GASES 


35 


Pressure  in  Close  Vessels,  Deduced  from  Theoretical  Con- 
siderations. —  The  expression  for  the  pressure  of  the  gases 
developed  by  the  combustion  of  gunpowder  in  a  close  vessel 
is  deduced  upon  the  following  suppositions  : 

i  st.  That  a  portion  of  the  products  of  combustion  is  in  a 
liquid  state. 

2d.  That  the  pressure  due  to  the  permanent  gases  can  only 
be  calculated  by  deducting  the  volume  of  the  liquid  products 
from  the  volume  of  the  vessel. 

Upon  these  hypotheses  the  expression  for  the  pressure  may 
be  deduced  as  follows  : 

Let  A  B  C  D  be  a  section  of  a 
close  vessel  of  volume  v  in  which  a 
given  charge  of  powder  is  exploded. 

Let  A  E  F  D  represent  the  space 
(vi)  occupied  by  the  charge,  and  A  G 
H  D  the  space  (v2)  occupied  by  the 
non-gaseous  products.  Let  At  be  the 
so-called  density  of  the  products  of 

combustion,  —  that  is  AI  =  --  ;   and 


B 


OL  the  ratio  of  the  non-gaseous  prod- 
ucts to  the  volume  of  the  charge,  or 


a  = 


V, 


—2-.     The  gases  after  ex- 


v 


D 


plosion  will  occupy  the  space  v  —  v2=  v  —  «  AI  z;  =  v  (i  —  a  AI). 
Let  pi  be  the  pressure  that  would  be  developed  if  the  volume 
of  the  vessel  were  A  E  F  D  (or  z;,).  In  this  case  the  density 
of  the  products  of  combustion  (At)  (the  charge  remaining  the 
same)  would  be  unity;  and  the  space  occupied  by  the  gases 
would  be  Vi  —  v2  =  Vi  (i  —  a)  =  At  v  (i  —  «).  Now  if  p  is  the 
pressure  when  the  volume  of  the  vessel  is  t>,  we  have  by  Mariotte's 
law  (assuming  that  the  temperature  is  the  same  for  all  densities 
of  the  products  of  combustion), 


36  INTERIOR   BALLISTICS 

AIT^(I— ar) 

or,  making 
we  have 


I  -  a 


The  factor  /is  called  the  force  of  the  powder. 

Value  of  the  Ratio  «.  —  Let  p2  and  ps  be  the  pressures  in 
the  same  vessel  produced  by  two  different  charges,  and  A2  and 
A3  the  corresponding  densities  of  the  products  of  combustion. 
Then  from  equation  (23)  (assuming/  to  be  the  same  for  all  values 

of  AO, 

A-, 


and 

whence  by  division, 


I-«A2       ^3A2* 
Therefore 

3A2-/>2A3) 


= 


A    A 


by  means  of  which  the  mean  value  of  a  can  be  determined  when 
a  sufficient  number  of  pressures,  corresponding  to  different 
values  of  AI,  have  been  found  by  experiment.  The  value  of  « 
finally  adopted  by  Noble  and  Abel  is  0.57. 

Determination  of  the  Force  of  the  Powder.  —  To  determine/ 
we  have  from  equation  (23), 

i  —.=57  A, 
f-t-     A         ; 

**1 

from  which  /  may  be  found  by  means  of  a  single  measured 
pressure  corresponding  to  a  given  density  of  the  products  of 


PROPERTIES  OF  PERFECT  GASES  37 

combustion.  When  At  =  i,  that  is,  when  the  vessel  is  completely 
filled  by  the  charge,  p  was  found  to  be  43  tons  per  square  inch, 
and  therefore  /  =  43  (i  —  .57)  =  18.49  tons  or  41417.6  pounds 
per  square  inch.  Therefore  Noble  and  Abel's  formula  for  the 
pressure  in  a  close  vessel  is,  for  different  densities  of  the  products 
of  combustion, 

A! 
p  =  18.49  ~~         —  r  tons  per  sq.  in. 

~t 

=  41417.6  -         -—  Ibs.  per  sq.  in. 
1  ~  -57  **i 

To  transform  this  equation  so  that  it  shall  express  the  press- 
ure in  kilos  per  dm.2  we  may  employ  a  simple  rule  which,  as  it  is 
of  frequent  use,  is  here  inserted  for  convenience  : 

RULE:  —  To  reduce  a  pressure  expressed  in  tons  per  square 
inch  to  the  same  pressure  expressed  in  kilos  per  dm.2,  add  to  the 
logarithm  of  the  former  the  constant  logarithm  4.1972544  and 
the  sum  is  the  logarithm  of  the  pressure  required. 

If  the  pressure  to  be  reduced  is  in  pounds  per  in.2  then  the 
constant  logarithm  to  be  added  is  0.8470064. 

Applying  this  rule  the  expression  for  the  pressure  of  the 
products  of  combustion  of  a  charge  of  gunpowder  fired  in  a  close 
vessel  is  found  to  be 

p  =  291200 


-.57     ! 

.  '  .  /  —  291200  kilos  per  dm.2 

It  will  be  seen  from  the  definition  given  to  At  that  it  is  the 
density  of  loading  as  defined  in  Chapter  III  when  the  gravi- 
metric density  of  the  powder  is  unity,  —  that  is,  when  a  kilo  of 
the  powder  fills  a  volume  of  a  dm.3;  or,  what  is  the  same  thing, 
when  a  pound  occupies  a  volume  of  27.68  cubic  inches;  and  in 
this  case,  when  At  is  unity  the  charge  just  fills  the  receptacle. 
Noble  and  Abel  were  careful  to  keep  the  gravimetric  density 
of  the  powder  they  experimented  with  as  near  unity  as  possible. 


38  INTERIOR   BALLISTICS 

Interpretation  of  f  .  —  It  will  be  seen  from  Equation  (23)  that 
the  quantity  designated  by  /  is  the  pressure  of  the  gases  when 


(i-  a 


that  is,  when  the  space  occupied  by  the  gases  is  equal  to  the 
volume  of  the  charge,  which  requires  that  the  vessel  should  have 
i  -j-  OL  units  of  volume.  Thus  if  the  kilogramme  and  litre  are  the 
units  of  weight  and  volume,  respectively,  the  volume  of  the 
vessel  must  be  1.57  litres  in  order  that  the  gases  may  occupy  a 
volume  of  one  litre,  and  have  a  tension  equal  to  /.  From  this 
/  may  be  denned  to  be  the  pressure  of  the  gases  of  unit  weight  of 
powder  occupying  unit  volume  at  the  temperature  of  combustion 

r,. 

If  e  is  the  weight  of  gas  furnished  by  the  combustion  of  unit 
weight  of  powder  we  have  from  Equation  (2), 

p!  1)i    =    €  R  Tij 

and  if  Vi  is  the  unit  of  volume,  there  results 

pl=f=€RT1     .....     (25) 

If  the  pound  is  the  unit  of  weight  the  unit  of  volume  is  27.68 
cubic  inches.  In  this  case  the  definition  of  /  requires  that  the 
volume  of  the  vessel  should  be  1.57  X  27.68  =  43459  cubic 
inches. 

The  value  of  e,  according  to  Noble  and  Abel,  is  0.43;  and 
therefore  the  pressure  of  unit  weight  of  the  gases  of  fired  gun- 
powder at  temperature  TI  is 


0-43* 

From  this  it  follows  that  the  pressure  of  one  pound  of  the  gases 
of  fired  gunpowder  at  temperature  of  combustion,  confined  in  a 
volume  of  27.68  cubic  inches,  is 

41417-6 

-  =  96320  Ibs.  per  square  inch. 


PROPERTIES  OF  PERFECT  GASES  39 

Also,  the  pressure  of  one  pound  of  the  gases  of  the  paragraph 
immediately  preceding,  confined  in  a  volume  of  one  cubic  foot, 
is,  in  pounds  per  square  foot, 

06320  X  27.68 

— —  =  222180  &g. 

12 

If  the  gravimetric  density  of  the  powder  be  unity,  and  y  and 
v  be  taken  in  pounds  and  cubic  inches,  respectively,  then  Equa- 
tion (23)  becomes 


Solving  with  reference  to  y  and  to  v  gives 

pv 


27.68  (a  ; 
and 

v  =  27.6&y(aP+f)  (2g) 

These  equations  are  useful  in  questions  involving  the  bursting 
of  shells,  etc. 

Theoretical  Determination  of  the  Temperature  of  Explosion 
of  Gunpowder. — Having  determined  the  value  of  /  from  the 
experiments,  we  can  deduce  the  temperature  of  explosion  by 
means  of  the  formula 

T!         2?3    f 

According  to  Noble  and  Abel's  experiments,  if  the  gravi- 
metric density  of  the  powder  is  such  that  a  kilogramme  occupies 
one  litre,  the  gases  furnished  by  its  combustion  will  fill  a  volume 
of  280  litres  at  o°  C.  under  the  normal  atmospheric  pressure  of 
103.33  kgs.  per  square  decimetre.  We  therefore  have 

280 

v°  ~  y  * 

and 

Po  =  I03-33 


40  INTERIOR   BALLISTICS 

whence 

273  X  291200 

C' 


>        103.33  X  280 

This  is  the  absolute  temperature  of  combustion  of  gunpowder 
according  to  Noble  and  Abel's  latest  deductions  from  their  ex- 
periments. Subtracting  273°  from  this  temperature  we  have 
temperature  of  explosion  =  2475°  C.  (4487°  F.). 

Mean  Specific  Heat  of  the  Products  of  Combustion.  —  From 
equation  (8),  we  have  when  W  =  o,  that  is,  when  no  external 
work  is  performed, 

Q  =  C,  (T,  -  273) 

in  which  Q  is  the  heat  of  combustion;  that  is,  the  quantity  of 
heat  that  unit  of  weight  of  the  explosive  substance  evolves,  under 
constant  volume,  when  the  final  temperature  of  the  products  of 
combustion  is  o°  C.  From  this  equation  we  find 


c 


-  273 


The  heat  of  combustion  was  determined  by  Noble  and  Abel 
in  the  following  manner: 

"A  charge  of  powder  was  weighed  and  placed  in  one  of  the 
smaller  cylinders,  which  was  kept  for  some  hours  in  a  room  of 
very  uniform  temperature.  When  the  apparatus  was  through- 
out of  the  same  temperature,  the  thermometer  was  read,  the 
cylinder  closed,  and  the  charge  exploded. 

1  '  Immediately  after  explosion  the  cylinder  was  placed  in  a 
calorimeter  containing  a  given  weight  of  water  at  a  measured 
temperature,  the  vessel  being  carefully  protected  from  radiation, 
and  its  calorific  value  in  water  having  been  previously  deter- 
mined. 

'  '  The  uniform  transmission  of  heat  through  the  entire  volume 
of  water  was  maintained  by  agitation  of  the  liquid,  and  the 
thermometer  was  read  every  five  minutes  until  the  maximum 


PROPERTIES  OF  PERFECT  GASES  41 

was  reached.  The  observations  were  then  continued  for  an  equal 
time  to  determine  the  loss  of  heat  in  the  calorimeter  due  to 
radiation,  etc.;  the  amount  so  determined  was  added  to  the 
maximum  temperature." 

In  this  way  the  heat  of  combustion  of  R.  L.  G.  and  F.  G. 
powders  was  found  to  be  705  heat-units;  that  is,  the  combustion 
of  a  unit  weight  of  the  powder  liberated  sufficient  heat  to  raise 
the  temperature  of  705  unit-  weights  of  water  i°  C.  We  there- 
fore have 


This  result  is  accepted  by  Noble  and  Abel,  and  also  by  Sarrau, 
as  a  very  close  approximation  to  the  mean  specific  heat  of  the 
entire  products  of  combustion.  If  we  assume  that  the  mean 
specific  heat  of  gunpowder  of  different  compositions  is  constant, 
we  can  compute  the  temperatures  of  combustion  when  the  heat 
of  combustion  has  been  determined  by  the  calorimeter,  by  the 
formula 

T          Q 

0.285 

in  which  T  is  given  by  the  centigrade  scale. 

Pressure  in  the  Bores  of  Guns  Derived  from  Theoretical 
Considerations.  —  "At  an  early  stage  in  our  researches,  when  we 
found,  contrary  to  our  expectation,  that  the  elastic  pressure  de- 
duced from  experiments  in  close  vessels  did  not  differ  greatly 
(where  the  powder  might  be  considered  entirely  consumed,  or 
nearly  so)  from  those  deduced  from  experiments  in  the  bores  of 
guns  themselves,  we  came  to  the  conclusion  that  this  departure 
from  our  expectation  was  probably  due  to  the  heat  stored  up  in 
the  liquid  residue.  In  fact,  instead  of  the  expansion  of  the  per- 
manent gases  taking  place  without  addition  of  heat,  the  residue, 
in  the  finely  divided  state  in  which  it  must  be  on  the  ignition  of 
the  charge,  may  be  considered  a  source  of  heat  of  the  most  per- 


42  INTERIOR   BALLISTICS 

feet  character,  and  available  for  compensating  the  cooling  effect 
due  to  the  expansion  of  the  gases  on  production  of  work. 

"The  question,  then,  that  we  now  have  to  consider  is — What 
will  be  the  conditions  of  expansion  of  the  permanent  gases  when 
dilating  in  the  bore  of  a  gun  and  drawing  heat,  during  their  ex- 
pansion, from  the  non-gaseous  portions  in  a  very  finely  divided 
state?"* 

Let  ct  be  the  specific  heat  of  the  non-gaseous  portion  of  the 
charge,  which  we  can  assume,  without  material  error,  to  be  con- 
stant. We  shall  then  have  ct  d  T  for  the  elementary  quantity  of 
heat  yielded  to  the  gases  per  unit  of  weight  of  liquid  residue.  If 
there  are  wt  units  of  weight  of  liquid  residue  it  will  yield  to  the 
gases  wv  ct  d  T  units  of  heat;  and  if  there  are  w2  units  of  weight 
of  gas  we  shall  have  in  heat-units, 


in  which 


that  is,  £  is  the  ratio  between  the  weights  of  the  non-gaseous 
and  gaseous  portions  of  the  charge.  The  negative  sign  is  given 
to  the  second  member  because  T  decreases  while  q  increases. 

Substituting  the  above  value  of  d  q  in  Equation  (7),  it  be- 
comes 

-(Cv  +  pCl)dT  =C-^pdv  .     .     .     .     (29) 


and  this  combined  with  Equation  (6),  gives,  by  a  slight  reduction, 

-  (ft  c,  +  €„•)?£  -(fa+CJ^   .     .     .     (30) 

Since  Cp,  CVJ  Ci  and  p  are,  by  hypothesis,  constant  during  the 
expansion,  the  integration  of  Equation  (30)  between  the  limits  v2 

*  Noble  and  Abel,  Researches,  etc.,  page  98. 


PROPERTIES  OF  PERFECT  GASES 


43 


and  vs — the  former  being  the  initial  volume  occupied  by  the  per- 
manent gases  and  the  latter  their  volume  after  the  projectile  has 
been  displaced  by  a  distance  u,  gives 


in  which 


r  = 


Equation  (31),  it  will  be  seen,  becomes  identical  with  Equa- 
tion (15),  when  /?  =  o;  that  is,  when  there  is  no  liquid  residue. 

To  introduce  Vi  and  v,  that  is  the  volumes  occupied  by  the 
charge  and  the  entire  volume  in  the  rear  of  the  projectile,  into 
Equation  (30)  in  place  of  v2  and  z>3,  proceed  as  follows:  Let 
ABC  D 


a?! 


E  F  G  H 

A  C  EG  represent  the  chamber  of  the  gun,  which  we  will  suppose 
filled  with  powder  without  compression,  and  further  that  one 
pound  of  the  powder  fills  a  space  of  27.68  cubic  inches.  The 
gravimetric  density  and  density  of  loading  are  each  unity;  and 
if  Vi  is  the  volume  of  the  chamber,  it  follows  that 

Vi  =  27.68  w. 

co  being  the  weight  of  charge. 

Suppose  the  powder  to  be  entirely  consumed  before  the  pro- 
jectile moves  any  perceptible  distance;  and  that  the  non-gaseous 
products  fill  the  space  A  B  E  F,  whose  volume  is  a  z>,.  The  gases, 


44  INTERIOR   BALLISTICS 

therefore,  which  by  their  expansion  give  motion  to  the  projectile 
will  occupy  the  space  B  C  F  G  before  perceptible  motion  begins. 
The  volume  of  the  space  B  C  F  G  is  evidently  v2  =  vt  —  a  ^  = 
Vi  (i  —  a).  Let  D  H  be  the  base  of  the  projectile  after  it  has 
moved  a  distance  «;  and,  designating  the  volume  A  D  E  H  by  v, 
we  evidently  have  v3  =  v  —  a  vlt  Substituting  these  values  of 
v2  and  vs  in  Equation  (31)  gives 


In  this  equation  ^  is  the  pressure  produced  by  the  combustion 
of  a  charge  of  powder  in  a  close  vessel  when  the  density  of  load- 
ing is  unity.  The  values  of  the  constants  are  given  by  Noble  and 
Abel  as  follows:* 

pi  =  43  tons  per  square  inch 

a  =  0.57 

^  =  1-2957 
Cp  =  0.2324 
Cv  =  0.1762 

ct  =  0.45 

Vi  =   27.68  <o 

from  which  we  find  r  =  1.074.     Substituting  these  values  in  the 
expression  for  p  it  becomes 


. 
=  43        - 


which  gives  the  pressure  in  tons  per  square  inch. 
If,  as  in  a  close  vessel,  we  let 


then 


~   -57  A! 


Researches,   etc.,  page    167, 


PROPERTIES  OF  PERFECT  GASES  45 

A   j        /  r-°74 


REMARKS:  —  The  value  of  ft  =  1.2957,  adopted  by  Noble  and 
Abel,  gives  for  unit  weight,  Wi  =  0.5644  and  w2  =  0.4356,  while 
the  values  of  these  quantities  adopted  in  our  equations  are  0.57 
and  0.43,  respectively.  These  last-named  values  would  make 
P  =  1.3256. 

Noble  and  Abel's  values  of  the  specific  heats  of  the  permanent 
gases  of  combustion,  namely,  Cp  =  0.2324  and  Cv  =  0.1762,  make 
n  =  1.32;  while  for  perfect  gases,  as  has  been  shown,  n  =  1.4 
very  nearly. 

Table  of  Pressures.  —  In  the  following  table  of  pressures  the 
third  column  gives  the  pressures  in  the  bore  of  a  gun  correspond- 
ing to  the  values  of  AI  in  the  first  column.  They  were  computed 
by  Equation  (34)  upon  the  assumption  that  the  permanent  gases 
in  expanding,  and  thereby  doing  work,  borrow  heat  from  the  non- 
gaseous  residue;  and  also  that  the  combustion  is  complete  before 
the  projectile  has  moved  perceptibly;  and  finally  that  there  is  no 
conduction  of  heat  to  the  walls  of  the  gun.  The  tensions  in  the 
fifth  column  were  computed  by  Equation  (24)  and  agree  with 
Noble  and  Abel's  experiments.* 

Temperatures  of  Products  of  Combustion  in  Bores  of  Guns.  — 
The  temperature  in  the  bore  of  a  gun  during  the  expansion  of 
the  products  of  combustion,  may  be  determined  from  Equation 
(29),  which  replacing  R  by  its  value  from  Equation  (i),  becomes 

dT  C-C< 


pC/  +  Cv 

whence  integrating  between  the  same  limits  as  before,  and  ob- 
serving that 


7T~7T^  =  r  -  i, 

'V 


*  For  table  of  pressures  see  page  46. 


46 


INTERIOR   BALLISTICS 


TABLE  OF    PRESSURES. 


f 

Mean  density 
of  products 
of  combustion. 

\ 

Corre- 
sponding ex- 
pansions. 

I 

"A^ 

Tensions  calculated  by 
Equation  (33). 

Tensions  in  close  cylinders, 
or  where  gases  expand 
without  doing  work. 

Tons  per  square 
inch. 

Differences. 

Tons  per  square 
inch. 

Differences. 

1  .00 

.000 

43-oo 

5.01 

43.00 

4.69 

•95 

•053 

37-99 

4.40 

38.31 

4.14 

.90 

.in 

33-59 

3-88 

34-17 

3.68 

•85 

.176 

29.72 

3-44 

30-49 

3-30 

.80 

.250 

26.28 

3-06 

27.19 

2.97 

•75 

•333 

23.22 

2.76 

24.22 

2.68 

.70 

.429 

20.46 

2.48 

21.54 

2-45 

•65 

•539 

17.98 

2.25 

19.09 

2.23 

.60 

.667 

15-73 

2.04 

16.86 

2.05 

•55 

.818 

13.69 

.86 

14.81 

1.88 

•50 

2.000 

11.83 

.70 

12.93 

i-74 

•45 

2  .222 

10.13 

-56 

II  .19 

1.61 

.40 

2.500 

8-57 

-43 

9-58 

•50 

•35 

2.857 

7-H 

•31 

8.08 

-39 

•30 

3-333 

5-83 

.21 

6.69 

30 

•25 

4.000 

4.62 

I  .11 

5-39 

.22 

.20 

5  .  ooo 

3-51 

I  .02 

4-1? 

H 

•15 

6.667 

2.49 

•93 

3-03 

.07 

.10 

10.000 

1-56 

.84 

1.96 

.01 

•  05 

20  .  ooo     !           .  72 

-95 

.... 

we  have 


Ti  l-r 


r  —  i 


Replacing  v2  by  Vi  (i  —  a),  and  vs  by  v  —  «  vi}  for  reasons 
already  given,  we  have  for  the  absolute  temperature  of  the  gases 
during  expansion,  the  equation 


T  . 


-  a\r  -  i 


—  a 


Introducing  the  density  of  the  products  of  combustion  (Ai), 
and  the  numerical  values  of  «  and  r  into  this  last  equation,  it 
becomes 


PROPERTIES  OF  PERFECT  GASES  47 


=  0.93946  7\  |  i  _   l    A~  |         ...     (35) 

The  value  of  T  for  any  given  density  of  the  products  of  com- 
bustion (represented  by  Ax)  will  depend  upon  their  initial  tem- 
perature (or  absolute  temperature  of  combustion),  Tv.  Its 
theoretical  value,  based  upon  Noble  and  Abel's  latest  deductions 
from  their  experiments,  as  published  in  their  second  memoir, 
has  already  been  found  to  be  2748°  C.  But  there  are  very  great 
difficulties  in  the  way  of  verifying  by  experiment  the  theoretical 
value  of  TI  ,  and  Captain  Noble  in  his  Greenock  lecture  (February 
1 2th,  1892)  takes  the  absolute  temperature  of  combustion  at 
2505°  C.,  as  deduced  in  their  first  memoir.  Therefore  making 

r,  =  2505°  c., 

the  expression  for  T  becomes 

r  =  2353-3JI_Al57Ai['°74      •      •      •     (36) 

The  temperatures  in  degrees  Centigrade  and  Fahrenheit,  cal- 
culated from  Equation  (36),  are  given  in  the  following  table. 
"It  is  hardly  necessary  to  point  out  that  the  values  given  in  this 
table  are  only  strictly  accurate  when  the  charge  is  ignited  before 
the  projectile  is  sensibly  moved;  but  in  practice  the  correction 
due  to  this  cause  will  not  be  great." 

Theoretical  Work  Effected  by  Gunpowder.— The  theoretical 
work  which  a  charge  of  gunpowder  is  capable  of  effecting  during 
the  expansion  of  its  volume  from  ^  to  any  volume  v  is  expressed 
by  the  definite  integral 


W  =          pdv, 


For  table  of  temperatures  see  next  page. 


48 


INTERIOR   BALLISTICS 
TABLE   OF   TEMPERATURES. 


Mean  density  of  prod- 
ucts of  combustion. 

A! 

Number  of  volumes  of 
expansion. 

I 
Al 

TEMPERATURES. 

Centigrade. 

Fahrenheit. 

I.OO 

.0000 

2231 

4048 

.95 

.0526 

2210 

4010 

.90 

.1111 

2189 

3972 

.85 

•  1765 

2168 

3934 

.80 

.2500 

2147 

3897 

.75 

•3333 

2126 

3859 

.70 

.4286 

2106 

3823 

.65 

.5385 

2085 

3785 

.60 

.6667 

2063 

3745 

•55 

.8182 

2041 

3706 

•50 

2  .  OOOO 

2018 

3664  ' 

•45 

2.2222 

1994 

3621 

.40 

2  .  5000 

1968 

3574 

•35 

2.8571 

1940 

3524 

•30 

3-3333 

1909 

3468 

•25 

4  .  oooo 

I874 

34°5 

.20 

5  .  oooo 

1834 

3333 

•15 

6.6667 

1785 

3245 

.10 

10.0000 

1719 

3126 

•05 

20  .  OOOO 

1615 

2939 

.00 

00 

0 

0 

or,  substituting  for  p  its  value  from  (32), 


v  r    dv 

=  pi  vAl   —  a)    I    -,  -  ^-', 

'  «A>!  (v  —  aVj)" 
whence,  integrating,  we  have 


W  =  ^-i 


-.    r  —  i 

Multiplying  and  dividing  the  second  member  by  hi  (i  —  «)]'  ', 
we  have 


(r  - 


PROPERTIES  OF  PERFECT  GASES  49 

If,  in  this  last  equation,  pi  be  expressed  in  kilogrammes  per 
square  decimetre,  and  Vi  be  made  unity  (one  litre),  the  work  will 
be  expressed  in  decimetre-kilogrammes  per  kilogramme  of 
powder  burned.  To  express  the  work  in  foot-tons  per  pound  of 
powder  burned,  we  must  make  Vi  =  27.68  cubic  inches;  and 
then,  since  pi  is  given  in  tons  per  square  inch,  divide  the  result 
by  12,  the  number  of  inches  in  a  foot.  Making  these  substitu- 
tions and  replacing  <*  and  r  by  their  values  already  given,  we 
have,  in  foot-tons, 

(  /     43^    N1-1 

W  =  576.369    i  -   L     ' 

•  D  / 

or,  in  terms  of  A1? 


i 

=  576.369!  i  -  0.93946  (-^—^)     j"     •    (37) 

Substituting  in  Equation  (37)  from  Equation  (35)  we  have 


or,  since,  according  to  Noble  and  Abel, 

ro 
i  =  25°5 

we  have 

W  =  0.23008  (Tt-T)      .....     (38) 

which  gives  the  work  in  terms  of  the  loss  of  temperature  of  the 
products  of  combustion. 

Table  III  gives  the  work  of  expansion  of  the  gases  of  one 
pound  of  gunpowder  of  the  normal  type  and  free  from  moisture, 
computed  by  Equation  (37)  .  By  means  of  the  work  given  in  this 
table,  and  by  the  use  of  a  proper  factor  of  effect  determined  by 
experiment,  Noble  and  Abel  consider  that  the  actual  work  of  a 
given  charge  of  powder  upon  a  projectile  may  be  computed  with 
considerable  accuracy.  Their  method  of  using  this  table  will  be 
clearly  seen  by  the  following  extract: 

"If  we  wish  to  know  the  maximum  work  of  a  given  charge, 

4 


50  INTERIOR   BALLISTICS 

fired  in  a  gun  with  such  capacity  of  bore  that  the  charge  suffered 
five  expansions  (A!  =  0.2)  during  the  motion  of  the  projectile  in 
the  gun,  the  density  of  loading  being  unity,  the  table  shows  us 
that  for  every  pound  in  the  charge,  an  energy  of  91.4  foot- tons 
will  as  a  maximum  be  generated. 

"If  the  factor  of  effect  for  the  powder  and  gun  be  known,  the 
above  values,  multiplied  by  that  factor,  will  give  the  energy  per 
pound  that  may  be  expected  to  be  realized  in  the  projectile. 

"But  it  rarely  happens,  especially  with  the  very  large  charges 
used  in  the  most  recent  guns  that  densities  of  loading  so  high  as 
unity  are  employed;  and  in  such  cases,  from  the  total  energy 
realizable  must  be  deducted  the  energy  which  the  powder  would 
have  generated,  had  it  expanded  from  a  density  of  unity  to  that 
actually  occupied  by  the  charge.  Thus  in  the  example  above 
given,  if  we  suppose  the  charge  instead  of  a  density  of  loading 
of  unity  to  have  a  density  of  0.8,  we  see  from  Table  3,  that  from 
the  91.4  foot-tons  above  given,  there  must  be  subtracted  19.23 
foot- tons;  leaving  72.17  foot- tons  as  the  maximum  energy  realiz- 
able under  the  given  conditions,  per  pound  of  the  charge." 

To  apply  these  principles  practically  for  muzzle  "velocities,  let, 
as  before, 

Vi  be  the  volume  occupied  by  the  charge,  in  cubic  inches. 
v  the  total  volume  of  bore  and  chamber,  in  cubic  inches. 
Vb  the  volume  of  the  bore. 
Vc  the  volume  of  the  chamber,  in  cubic  inches. 
Then 

v=Vb+Vc; 

and,  if  the  gravimetric  density  of  the  powder  be  unity, 

Vi   =   27.68  co, 

where  co  is  the  weight  of  the  charge  in  pounds.     Therefore  the 

*  Noble  and  Abel,  Researches,  page  176. 


PROPERTIES  OF  PERFECT  GASES  51 

number  of  volumes  of  expansion  of  the  products  of  combustion 
will  be,  at  the  muzzle, 


JLs.i     _A_  ,i. 

l\     ~    A!        27.68  co         "V 


which  may  be  written,  if  the  gravimetric  density  of  the  powder 
be  unity, 

—  =  0.0361263-^+  — 

A!  u  A 

in  which  A  is  the  density  of  loading  as  denned  in  Chapter  III. 

If  the  gravimetric  density  of  the  powder  be  not  unity,  let  v2 
be  the  volume  in  cubic  inches  of  one  pound  of  powder  not  pressed 
together  except  by  its  own  weight;  and  let 

27.68 

—  -  —  =  m\ 

^ 

then  we  have  in  all  cases, 

^  =  m\  0.0361263^+^ 

in  which  —  is  the  number  of  volumes  of  expansion  of  the  prod- 

ucts of  combustion. 

Let  W2  be  the  work  taken  from  Noble  and  Abel's  table  (Table 
III)  of  the  gases  of  one  pound  of  powder  for  a  given  value  of 

—  ,  and  Wi  the  work  due  to  the  expansion  —  -.     Also,    let   F 

be  the  factor  of  effect.  Then  if  we  assume  that  the  work  of 
expansion  is  all  expressed  in  the  energy  of  translation  of  the 
projectile,  we  shall  have  approximately, 

-FW*  ......  (39) 

in  which  w  is  the  weight  of  the  projectile  and 

W  =  W2  -  Wl 
From  (39)  the  muzzle  velocity  v  may  be  computed  when  the 


52  INTERIOR   BALLISTICS 

factor  of  effect  is  known;  or,  we  may  determine  the  factor  of 
effect  when  the  muzzle  velocity  has  been  measured  by  a  chrono- 
graph. These  two  equations  reduced  to  practical  forms  are  the 
following : 

»  =  379-57  \IFW-     ....     (40) 
\  w 

and 


F  =  0.000006041  -fjT —     .      .      .      .      (41) 
w  & 

As  an  illustrative  application  of  these  formulas  to  interior 
ballistics  take  the  following  data  from  Noble  and  Abel's  second 
memoir,  relative  to  the  English  8-inch  gun:  It  was  found  by 
firing  a  charge  of  70  pounds  of  a  certain  brand  of  pebble  powder, 
with  a  projectile  weighing  180  pounds,  that  a  muzzle  velocity  of 
1694  foot-seconds  was  obtained.  What  was  the  factor  of  effect 
(F)  pertaining  to  this  gun  and  brand  of  powder  ?  For  this 
particular  gun  and  charge  we  have  w  =  70  pounds,  w  =  180 
pounds,  A!  =  0.1634,  A  =  0.605  andw  =  i.  In  Noble  and  Abel's 

table  of  work  (Table  III)   the  first  column  gives  values  of  — , 

increasing  by  a  common  difference,  while  the  second  column  con- 
tains the  corresponding  values  of  At.  By  a  simple  interpolation 
we  find  for  the  values  of  At  and  A  given  above,  W2  =  99.4  and 
Wi  =  37.6;  whence  W  =  61.8  foot-tons.  Substituting  these 
values  in  Equation  (41)  we  have 

1 80  X  (i694)2 

F  =  0.000006041 2 — 5~~  =  0.8287. 

70  X  61.08 

That  is,  the  actual  work  realized,  as  expressed  and  measured 
by  the  projectile's  energy  of  translation,  as  it  emerges  from  the 
bore,  is  nearly  83  per  cent,  of  the  theoretical  maximum  work 
which  the  powder  gases  are  capable  of  performing,  leaving  but 
17  per  cent,  for  the  other  work  done  by  the  gases,  namely,  the 
work  expended  upon  the  charge,  the  gun  and  carriage,  and  in 


PROPERTIES  OF  PERFECT  GASES  53 

giving  rotation  to  the  projectile;  the  work  expended  in  overcom- 
ing passive  resistances,  such  as  forcing  the  rotating  band  into 
the  groove,  the  subsequent  friction  as  the  projectile  moves  along 
the  bore,  and  the  resistance  of  the  air  in  front  of  the  projectile; 
and  lastly,  the  heat  communicated  to  the  walls  of  the  gun.  It  is 
very  difficult  to  evaluate  these  non-useful  energies,  but  it  is  prob- 
able that  they  do  not  consume  more  than  17  per  cent,  of  the 
maximum  work  of  the  gases.  Longridge  finds  by  an  elaborate 
calculation  that  this  lost  work  in  a  lo-inch  B.  L.  Woolwich  gun 
amounts  to  30  per  cent,  of  the  maximum  work;  *  but  it  is  believed 
that  he  has  greatly  overestimated  the  work  required  to  give 
motion  to  the  products  of  combustion.  Colonel  Pashkievitsch 
makes  the  lost  work  rather  less  than  17  per  cent,  of  that  expressed 
in  the  energy  of  translation  of  the  projectile.! 

To  test  the  correctness  of  Equation  (40)  for  determining 
muzzle  velocities  we  will  apply  it  to  the  same  gun  by  means  of 
which  the  factor  of  effect  was  determined,  increasing  the  charge 
from  70  to  90  pounds,  and  again  to  100  pounds,  and  compare  the 
computed  velocities  with  those  measured  with  a  chronograph. 
For  a  charge  of  90  pounds  of  powder  we  have  A!  =  0.210  and 
A  =  0.780;  whence  W2  =  89.3,  Wi=  20.86,  and  W  =  68.44 


1.8287X90X68.44  =  2Q2i  f 
'\  180 

The  measured  velocity  with  this  charge  was  2027  foot-seconds. 
In  a  similar  way  we  find  by  the  formula  that  for  a  charge  of  100 
pounds  v  =  2174  foot-seconds,  while  the  measured  velocity  was 
2182  foot-seconds.  The  differences  between  the  computed  and 
observed  velocities  in  these  examples  are  about  one-third  of  one 


*  "  Internal  Ballistics."  By  Atkinson  Longridge.  London,  1889. 
Chapter  V. 

f  "  Interior  Ballistics.  "  By  Colonel  Pashkievitsch.  Translated  from 
the  Russian  by  Captain  Tasker  H.  Bliss,  U.  S.  Army.  Washington,  1892. 


54  INTERIOR   BALLISTICS 

per  cent.,  and  are  well  within  the  limits  of  probable  error  in 
measuring  them. 

The  factor  of  effect  increases  with  the  caliber  of  the  gun,  as  is 
shown  by  experiment.  Thus  with  the  English  lo-inch  gun  fired 
with  charges  of  130  and  140  pounds  of  the  pebble  powder  we 
have  been  considering,  the  factor  of  effect  is  0.855;  while  with 
the  n-inch  gun,  and  charge  of  235  pounds,  the  factor  of  effect 
is  0.89. 


CHAPTER    III 

COMBUSTION  UNDER  CONSTANT  PRESSURE 

Combustion  of  a  Grain  of  Powder  Under  Constant  Atmos- 
pheric Pressure. — In  what  follows  it  is  assumed  that  the  powder 
grain  is  of  some  regular  geometrical  form  to  which  the  elementary 
rules  of  mensuration  can  be  applied.  It  will  also  be  assumed  as 
the  result  of  observation,  that  the  combustion  of  the  grain  takes 
place  simultaneously  on  all  sides  and  that,  under  the  constant 
pressure  of  the  atmosphere,  parallel  layers  of  equal  thickness  are 
burned  away  in  equal  successive  intervals  of  time — that  is,  that 
the  velocity  of  combustion  under  constant  pressure  is  uniform. 

The  form  and  dimension  of  each  grain  of  powder  constituting 
the  charge  are  of  the  utmost  importance,  as  upon  them  depends 
the  proper  distribution  of  the  mean  effective  pressure  within 
the  bore.  If  the  initial  surface  of  combustion  of  the  charge  be 
large  and  the  web  thickness  of  the  grains  small,  then  the  maxi- 
mum pressure  will  be  excessive  and  the  muzzle  velocity  inade- 
quate. On  the  other  hand,  if  the  web  thickness  be  too  great 
the  chase  pressure  may  prove  destructive  to  the  gun.  More 
than  one  of  our  heavy  guns  it  is  believed  have  been  wrecked 
during  the  past  ten  years  simply  from  excessive  web  thickness. 

Many  forms  of  grain  have  been  adopted  by  different  manu- 
facturers in  this  and  foreign  countries,  but  they  may  all  be 
divided  into  two  general  groups,  viz.:  those  burning  with  a 
continuously  decreasing  surface,  and  those  in  which  the  surface 
of  combustion  may  increase  (or  decrease)  to  a  certain  stage,  the 
grain  then  breaking  up  into  other  forms  entirely  dissimilar  to 
the  original  and  which  are  then  consumed  with  a  rapidly  de- 
creasing surface.  To  the  first  group  belong  spherical,  cubical, 
ribbon-shaped,  and  indeed  all  solid  grains  of  whatever  form, 

55 


56  INTERIOR   BALLISTICS 

and  cylindrical  grains  with  an  axial  perforation.  To  the  latter 
group  belong  pierced  prismatic  and  the  so-called  multiperforated 
grains  employed  by  both  our  army  and  navy. 

Notation. — Let 

/    =  thickness  of  layer  burned  in  time  /. 

10  =  one-half  the  least  dimension  of  the  grain.  Since  com- 
bustion takes  place  on  all  sides  of  a  grain  at  once,  it  may  be 
assumed  that  when  /  =  10  all  grains  of  the  first  group  are  totally 
consumed.  This,  of  course,  is  not  the  case  with  m.  p.  grains. 

S0  =  the  total  initial  surface  of  combustion  of  the  grain. 

S  =  surface  of  combustion  at  time  t,  corresponding  to  /. 

S'  =  the  total  burning  surface  when  I  =  I0'y  that  is,  when  the 
grain,  as  a  grain,  is  about  to  disappear.  This  surface  may  be 
called  the  vanishing  surface  of  combustion. 

V0  =  the  initial  volume  of  a  grain. 

V  =  volume  of  grain  burned  at  time  t.  That  is,  the  volume 
comprised  between  the  surfaces  S0  and  5. 

V 
k  =  fraction  of  grain  burned  in  time  /.     That  is,  k  =  TT- 

*  o 

The  general  expression  for  the  burning  surface  of  a  grain  of 
powder  moulded  into  any  one  of  the  simple  geometrical  forms 
adopted  by  powder  manufacturers  may  take  the  form, 

S  =  S0  +  a  I  +  bF (i) 

where  /  is  the  thickness  of  layer  burned  from  instant  of 
ignition.  At  that  instant  /  is  zero  and  5  the  initial  surface  of 
combustion  S0.  In  the  course  of  burning  when  /  is  about  to 
become  10,  S  is  about  to  become  Sf.  Therefore 

S-'  =  S0  +  al0  +  bi: (2) 

In  these  two  equations  a  and  b  are  constants  for  the  same 
form  of  grain,  whose  values  will  be  deduced  later. 

The  general  expression  for  the  volume  consumed  while  a 
thickness  /  is  burned  away,  is 


COMBUSTION    UNDER    CONSTANT    PRESSURE  57 

whence  substituting  for  5  its  general  value  from  (i)  and  inte- 
grating, 


The  initial  volume  V0  is  evidently  what  V  becomes  when  the 
grain  is  completely  consumed,  that  is,  when  /  =  10.     Therefore 


This,  of  course,  gives  the  entire  original  volume  only  for 
those  grains  which  are  completely  consumed  when  I  =  10,  or,  in 
other  words,  when  the  web  thickness  is  burned.  It  need  hardly 
be  said  that  it  does  not  apply  to  m.  p.  grains.  In  this  latter 
case,  it  gives  the  original  volume  minus  the  "  slivers,"  so  called. 

If,  in  (4),  we  substitute  for  S0  its  value  from  (2),  namely, 

S0  =  S'  -  a  10  -  b  102 
it  becomes 

F.-S').-4V-^.'    ....     (5) 

o 

From  (4)  and  (5)  we  readily  find 

...     (6) 


and 

b  =j-,(S.  +  S')  --T^      ....     (7) 

^o  lo 

These  equations  give  a  and  b  when  S0,  S'  and  V0  can  be 
computed  by  the  rules  of  mensuration.  It  will  be  observed  that 
a  is  a  linear  quantity  while  b  is  of  zero  order  of  magnitude.  These 
properties  afford  tests,  as  far  as  they  go,  as  to  whether  the  work 
of  deducing  a  and  b  in  any  particular  case  has  been  correctly 
performed. 


58  INTERIOR   BALLISTICS 

Fraction  of  Grain  Burned  for  any  Value  of  1. — We  have  by 
definition 


k  =  _L  = 2_ _3 

This  may  be  transformed  into 


V0 


ju,  M       a_t  _L  ,  *v  L 

V    'I       -  " 


o       vo 

Put  for  convenience, 


Then 

7    (  /  72  ) 

....     (9) 

For  all  grains  of  the  first  group  k  becomes  unity  when  /  =  10, 
that  is,  when  the  grain  is  all  burned ;  in  this  case  (9)  reduces  to 

i  =  a  (i  +    \+  /*) (10) 

This  relation  always  subsists  between  these  numerical  con- 
stants and  serves  to  test  the  correctness  of  their  derivation  in 
any  case. 

The  following  relations  which  are  easily  established  will  be 
useful : 

aV0 

S'  =  (i  +  2\  +  3tiS0  -$.-f-jT£(aX+3J»);    (n) 

or,  more  generally, 


We  also  have 


a  (X   +  2  /.)  =  ~°-  -  i. 

'    n 


COMBUSTION     UNDER     CONSTANT     PRESSURE  59 

Therefore  for  all  grains  whose  vanishing  surface  (S')  is  zero, 
we  have 

I    +    2\+  3  fJ.   =    O. 

and 

a  (\+  2f*)  =-  i (12) 

Applications. — We  will  now  apply  these  formulas  to  a  dis- 
cussion of  various  forms  of  grain  now  in  use  or  which  may 
come  into  use. 

i.  Sphere. — For  a  spherical  grain  10  is  evidently  the  radius. 
Then  by  mensuration 

S0  =  4^lo 


Substituting  these  in  (6)  and  (7)  we  readily  find 

a  =  —  8  TC  10  and  b  =  4  n. 
Therefore  from  (i) 

and,  therefore,  5  is  a  decreasing  function  of  /. 
From  (8)  we  find 

a  =  3,  X  =  -  i  and  fJ.  =  -; 

o 

and  these  substituted  in  (9)  give 

/\3 


/     i  r-  j          /      i\ 

T  +  ~  71      =    I    -   (  I   -  T) 

In  S      "ft      I  x  I'n' 


which  is  the  fraction  of  grain  burned  in  terms  of  the  thickness 
of  the  layer  /. 

If  we  divide  the  thickness  of  web  (radius  of  grain)  into  five 
equal  parts  the  following  table  may  be  computed,  which  will  be 
useful  for  comparing  this  form  of  grain  with  others  to  be  given: 


6o 


INTERIOR   BALLISTICS 


/ 

I. 

k. 

First  Differences. 

0.0 

o.ooo 



O.2 

0.488 

0.488 

0.4 

0.784 

0.296 

0.6 

0.936 

0.152 

0.8 

0.992 

0.056 

I.O 

I.OOO 

O.OO8 

The  second  column  gives  the  entire  fraction  of  grain  burned 
and  the  third  column  the  fraction  of  grain  burned  for  each  layer. 
It  will  be  observed  that  nearly  one-half  the  initial  volume  of  the 
grain  is  in  the  first  layer. 

2.  Parallelopipedon. — Let  2  10  be  the  least  dimension  of  the 
parallelopipedon  and  m  and  n  the  other  two  dimensions.  Then, 
by  the  rules  of  mensuration, 

S0  =  4l0m  +  4l0n  +  2mn 

S'  =  2  (m  -  2  10)  (n  -  2  Q  =  2  m  n  -  4 10  m  -  4  10  n  +  8  /02 

V0=  2l0mn 

Substituting  these  values  of  S0,  Sf  and  V0  in  (6)  and  (7),  gives 

a  =  —  8  (2  10  +  m  -\-  n)  and  b  =  24. 
Making  the  following  substitutions,  viz. : 


2  L 


2  I 


-  u  i  u 

-  =  x  and  —  =  y 
m  n 

in  which  x  and  y  are  generally  less  than  unity,  we  have,  finally, 

x  +  y  +  xy  ocy 

y 


It  may  be  noted  that  these  values  of  a,  X,  //  satisfy  equation  (10). 


COMBUSTION     UNDER    CONSTANT    PRESSURE  6  1 

There  are  three  special  parallelopipedons  worthy  of  separate 
notice  : 

(a)  Cube.  —  The   cubical  form  has  been  used  for  ballistite 
and  for  some  other  powders.     For  this  form  we  evidently  have 

x  =  y  =  i. 
Therefore 

a    =  3;  \   =    —   i;  p   =  -. 
o 

These  are  the  same  as  were  found  for  spherical  grains,  as 
might  have  been  inferred.  They  also  apply  approximately 
to  sphero-hexagonal,  mammoth  and  rifle  powders  (old  style). 

(b)  Square  Flat  Grains.  —  For  these  grains  (still  used  with 
certain  rapid-firing  guns),  m  and  n  are  equal  and  greater  than 
2  10.     Therefore  x  and  y  are  equal  and  less  than  unity.     There- 
fore, 


If  these  grains  are  very  thin,  x  becomes  a  very  small  fraction 
and  may  be  omitted  in  comparison  with  unity.  In  this  case  X 
and  IJL  are  approximately  zero  and  a  unity.  This  gives 


or,  a  constant  emission  of  gas  during  the  burning;  but  the  grain 
would  be  consumed  in  a  very  short  interval  of  time. 

(c]  Grains  Made  Into  Long  Slender  Strips  (or  "Ribbons"), 
with  Rectangular  Cross-Section.  —  These  grains  are  approximately 
those  of  the  new  English  powder  called  "axite."  Also  of  the 
French  "B  N"  powders,  and  others.  If  we  suppose  the  width 
of  the  strip  to  be  five  times,  and  the  length  one-hundred  and 
fifty  times,  its  thickness  (which  corresponds  nearly  with  the 

"  B  N  "  powders)  ,  we  shall  have  x  =  —  and  y  =  --  .     Therefore 

5 


a  =  1.207;  ^  =  —  0.172;  /z  =  o.ooi; 


62  INTERIOR   BALLISTICS 

and  the  expression  for  k  becomes 

I 


k    =    1. 2O7T--J    I  —0.172 

In    ( 


The  following  table  illustrates  the  progressiveness  of  this 
particular  grain : 


I 

lo 

k. 

First  Differences. 

0.0 

o.ooo 

0.2 

0.233 

0-233 

0.4 

0.450 

0.217 

0.6 

0.650 

0.200 

0.8 

0.833 

0.183 

I.O 

I.  000 

o.  167 

I.  000 

These  strips,  made  up  into  compact  bundles  or  fagots  to 
form  the  charge,  seem  well  adapted  for  rapid-firing  guns  of 
moderate  caliber.  In  the  application  of  the  expression  for 
k  for  computing  velocities  and  pressures  in  the  gun,  fj,  may  be 
regarded  as  zero,  and  thus  greatly  shorten  the  calculations  with- 
out impairing  their  accuracy. 

If  the  cross-section  of  the  strip  is  square,  we  shall  have 

2  I, 

m  =  2  10,  x  =  i  and  y  =  — , 

n  being  the  length  of  the  strip.     Therefore,  in  this  case, 

1  +  2  y  y 

a  =  2  +  y;  X  = ;  ft  =  -     — . 

2  +  y   '  2  +  y 

If  the  strip  be  very  long  in  comparison  with  the  linear 


COMBUSTION     UNDER    CONSTANT    PRESSURE  63 

dimension  of  cross-section,  y  may  be  considered  zero,  and  we 
have 

a  =  2;  X  =  --;/£  =  o. 
Therefore 

k    =    2-\I    ~--}     =    I    -(l          -^ 


3.  Solid  Cylinder. — For  this  form  of  grain  there  are  two 
cases  to  be  considered:  (a)  When  the  diameter  of  cross-section 
of  the  cylinder  is  the  least  dimension,  (b)  When  the  length  of 
the  cylinder  is  the  least  dimension.  That  is,  a  cylinder  proper 
and  a  circular  disk. 

(a)  Cylinder  Proper. — In  accordance  with  the  notation 
adopted,  10  will  be  the  radius  and  m  the  length  of  the  cylinder. 
We  have  by  mensuration, 

S0  =  2  TT  (10  m  +  /02);  Sf  =  o;   V0  =  *  102  m\ 
whence 

a  =  —  2  TT  (4  10  -f  m)  and  b  =  6  n. 

2  10 
Putting,  as  before,  — -  =  x,  there  results 


These  are  the  same  expressions  for  «,  X,  /*  as  was  found  for 
a  strip  with  square  cross-section,  as  might  readily  be  inferred. 

If  x  be  small  in  comparison  with  unity,  that  is,  if  the  grains 
are  long  slender  cylinders  (thread  like),  like  cordite,  we  have 
very  approximately, 

a  =  2;  X=  -  j;/*  =  o; 

and,  as  before  shown, 

IV 


k 


=•-(•-) 


64  INTERIOR   BALLISTICS 

The  following  table  was  computed  by  this  formula: 


J_ 

k. 

First  Differences. 

0.0 
O.2 

0.00 

0.36 

0.36 

0.4 

0.64 

0.28 

0.6 

0.84 

O.2O 

0.8 

0.96 

0.12 



0.04 

I.O 

I.  00 

I  .00 

Comparing  this  table  with  that  given  for  "strips,"  it  will 
be  seen  that  the  burning  of  cordite  is  not  so  progressive  as  that 
of  axite. 

If  the  length  of  the  solid  cylindrical  grain  be  the  same  as  its 
diameter,  then  x  =  i  ;  and  we  have 


as  for  spherical  and  cubical  grains. 

(b)  Circular  Disk.  —  With  this  form  of  grain  the  thickness 
becomes  the  least  dimension  instead  of  the  diameter.  Let  2  10 
be  the  thickness  of  the  disk  and  R  its  radius. 

Then 

S0  =  2  TT  R  (2  10  +  #>;  S'  =  2  K  (R  -  /0)2;  V0  =  2  n  10  R\ 
Whence 

a  =  —  4  TT  (2  R  +  10)  and  b  =  6  n. 


2  I 


Therefore  making  -^  =  ~  =  x,  we    have,  as    has    already 


COMBUSTION     UNDER     CONSTANT     PRESSURE  65 

been  found  for  square  flat  grains, 

x  (2  +  x)  x2 


a    =  i  +  2  #:  X  =   — 


I   +  2JC    '  I   +  20C 


4.  Cylinder  with  Axial  Perforation.  —  Let  R  =  radius  of 
grain,  r  =  radius  of  perforation,  and  m  =  its  length.  We  then 
have  i 

2l0  =  R-r, 
and 

R  +  r  =  2(R~10).     .  '  .  R2  -  r2  =  4  10  (R  -  Q. 

By  the  rules  of  mensuration,  we  find,  after  reduction, 

S0  =  2  TT  m  (R  +  r]  +  2  TT  (R-  r)  =4x(m  +  2  10)  (R  -  10\ 

sf  -4  *•(»-•«  Wit*  -4) 

V0=4*l0™(R-lo) 

Therefore 

a  =  —  16  T:  (7?  —  /0)  and  b  =  o. 

2  / 
Making,  as  before,  x  =  -  -  we  have 


Therefore 


As  an  example  of  this  form  of  grain,  suppose  the  length  to 
be  three  hundred  times  the  thickness  of  web.     Then 


x  =  -  ;  a  =  -  —  ;  X  =  —  -  —  ;  fi  =  o. 
300'          300'  301" 

The  expression  for  k  is 


_  _ 

300  /0  /       301  /0  5    ^  (  300     300  /0 


66  INTERIOR  BALLISTICS 

The  following  table  was  computed  by  this  formula : 


I 

c 

k. 

First  Differences. 

o  o 

O.QQOO 

O.2 

0.2005 

0.2005 

0.4 

0.4O08 

0.2003 

0.6 

0.6008 

O.2OOO 

0.8 

0.80O5 

0.1997 
O    IQQS 

I  .0 

I  .OOOO 

I.  OOOO 

This  form  of  grain  is  very  progressive,  much  more  so  than 
any  other  form  that  has  been  proposed,  and  seems  well  adapted 
for  guns  of  all  calibers.  The  first  differences  show  that  for  all 
practical  purposes  the  emission  of  gas  may  be  considered  constant 
during  the  entire  burning  of  the  grain. 

From  (n)  we  have,  when  /*  =  o, 


Therefore  in  this  example,  when  x  =  --  .we  have 

3°° 


and  the  burning  surface  during  the  entire  combustion  lies  be- 


tween  its  initial  value  S0  and  its  final  value  --  S0. 

\J 

5.  Multiperf  orated  Grains.  —  These  grains,  which  are  used 
exclusively  with  the  heavy  artillery  of  the  army  and  navy  of  the 
United  States,  are  cylindrical  in  form  and  have  seven  equal 
longitudinal  perforations,  one  of  which  coincides  with  the  axis 


COMBUSTION     UNDER    CONSTANT    PRESSURE  67 

of  the  grain,  while  the  others  are  disposed  symmetrically  about 
the  axis,  their  centres  joined  forming  a  regular  hexagon.  The 
web  thickness  (2  Q  is  the  distance  between  any  two  adjacent 
circumferences ;  and  therefore,  if  R  is  the  radius  of  the  grain  and 
r  the  radius  of  each  of  the  perforations,  we  have  the  relation 

2l    -    R~3r 

2 

From  the  geometry  of  the  grain  as  denned  above  we  have 
the  following  relations: 

S0=2r:  [F -7r>  +  m(R  +  7r)}  .     .     .     (13) 

S'  =  S0 +  4*lo(3™-  2(R  +  7r)  -910)  .     (14) 

V0  =  r.  m  (R2  -  7  r)  =  ™{S0  -  2  r,  m  (R  +  7  r)  }     (15) 
V'0  =  /0S0  +  27r/02(3m-  2(R  +  7r)  -610)       .     (16) 

In  these  expressions  S0  and  V0  are  the  initial  surface  of 
combustion  and  volume,  respectively,  while  Sf  and  V'0  are  the 
vanishing  surface  and  volume  burned,  when  /  is  about  to  become 
10  and  the  grain  to  break  up  into  slivers.  If  we  substitute  the 
values  of  S0,  S'  and  V'0  from  the  above  equations  in  (6)  and  (7), 
they  reduce  to 

a  =  4  TT  (3  m  -  2  (R  -f  7  r)) 
and 

b  =  -  36  TT 

These  values  of  a  and  b,  in  equations  (8),  give 


__      _  (    \ 

'  R2  -  7  r3  +  m  (R  +  7  r) 

These  values  of  a,  X,  and  ^  satisfy  the  equation  of  condition 

a  (i  +  X  +  ,«)  =  i, 


68  INTERIOR   BALLISTICS 

since  when  I  =  10  the  volume  V'0  has  been  consumed.  When 
this  occurs,  the  original  form  of  the  grain  disappears  and  there 
remain  twelve  slender,  three-cornered  pieces  with  curved  sides 
technically  called  "slivers."  These  of  course  must  be  treated 
differently.  In  the  applications  of  these  formulas  given  in 
Chapter  V,  the  form  characteristics  of  the  slivers  are  assumed 

to  be  OL  =  2,  X  =  —  and  /*  =  o,  with  good  results. 

The  form  characteristics  deduced  in  (17),  (i  8),  and  (19), 
if  substituted  in  (9),  will  give  the  fraction  of  volume  V  '  0  burned. 
But  what  is  required  in  practice  is  the  fraction  of  the  entire 
grain  (or  charge).  This  is  found  by  employing  V0  instead  of 
V0.  By  this  means  we  find 


and  this  value  of  OL  will  be  used  in  all  the  applications.  The 
expressions  for  A  and  /*,  being  independent  of  the  volume  (see 
equations  (8)),  are  those  deduced  above. 

Substituting  the  form  characteristics  in  (9)  and  making 

l  =  lo 

we  shall  have  the  fraction  of  the  entire  grain  burned  when  the 
web  thickness  is  burned.  Calling  this  fraction  kr  it  will  be 
found  that 

,_!.(          (n-^(R  +  ,r  +  3U 

m  (  R2  •-  7  r2  ) 

This  expression  for  k'  would  also  be  obtained  by  dividing 
(16)  by  (15). 

It  will  be  seen  that  for  the  same  web  thickness  a  and  k' 
decrease  as  m  increases,  but  within  moderate  limits,  their  limit- 
ing values,  when  m  is  infinite,  being 


R2  -  7  r3 


COMBUSTION    UNDER    CONSTANT    PRESSURE  69 

and 

,  _  2l0(R  +  jr+3l0) 
R2  -  7  r* 

For  the  grains  employed  in  the  United  States  service,  the 

D 

ratio  —  varies  but  little  from  n.     If  we  adopt  this  ratio,  the 
expressions  for  the  form  characteristics  a,  X,  /*  and  k'  become 

12          2/0 

19   '    m 
2(m-6 10 


X  = 


19/0  +  6  w 


6m 


k>  = 


19     19  m 


We  also  have  R  =  5.5  10  and  r  =  0.5  10. 

If,  in  addition,  we  make  m  =  n  10  we  have 

12        2 

<*  = 1 

19        n 

a  (»  -  6) 

6w  +  19 

4 


M     =    - 


+ 19 

6 


19       19  n 

It  may  be  noted  that  the  limiting  values  of  these  form 
characteristics,  as  the  length  of  the  grain  is  indefinitely  increased, 
are, 

a  =  — ;  X  = — ;  /z  =  o  and  kf  =  — . 
J9  3  19 


70  INTERIOR   BALLISTICS 

Also  that  X  is  zero  when  n  =  6  and  becomes  negative  when  the 
grain  is  still  further  shortened. 

It  will  be  seen  that  the  percentage  of  slivers  can  never  be 
greater  than  about  16. 

For  the  grains  in  use  n  is  approximately  26,  which  gives 

a  =  —  =  0.70850 
247  /     * 

8 

X  =  —  =  0.22857 
35 

/£  =    -  -~^  =    -  0.022857 

*'  =  ^7  =  °-85425 

There  seems  to  be  no  valid  reason  why  these,  or  other  simple 
ratios,  for  R/r  and  m/l0  should  not  be  adopted  by  powder 
manufacturers  for  all  sizes  of  m.p.  grains,  making  the  diameter 
of  the  grain  and  perforations,  and  also  its  length,  depend  upon 
the  web  thickness  adopted  for  a  particular  gun.  For  example, 
the  web  thickness  adopted  for  the  i4-inch  gun  is  0.1454  inch. 
Therefore  the  dimensions  of  the  grains  would  be 

Diameter  =  5.5  X  0.1454  =  0.7997  in. 
Diameter  of  perforation  =  0.1454/2  =  0.0727  in. 
Length  =  13  X  0.1454  =  1.89  in. 

These  dimensions  are  practically  the  same  as  those  of  the 
actual  grains.  From  eqjation  (26')  of  this  chapter  it  will  be 
seen  that  the  initial  surface  of  one  pound  of  these  grains  would 
vary  inversely  as  the  web  thickness. 

For  these  grains,  equations  (13)  to  (16)  reduce  to 

S0  =  $2$*l* 

S'=   729  *l? 

V0   =    741  *  lo 


COMBUSTION    UNDER    CONSTANT    PRESSURE  71 

The  vanishing  surface  is  therefore  about  39  per  cent,  greater 
than  the  initial  surface. 

Captain  Hamilton  has  shown  conclusively  that  the  m.p. 
grains  now  in  use  are  much  too  short  to  secure  a  proper  alignment 
in  the  powder  chamber,  and  that  this  lack  of  alignment  conduces 
to  excessive  pressure.* 

If  we  make  n  =  200,  that  is,  make  the  length  of  the  grains 
100  times  the  web  thickness,  we  should  have 

a  =  0.64158 
X  =  0.31829 
/z  =  —  0.00328 
k'  =  0.84368 

This  value  of  n  would  make  the  length  of  the  grains  for  the 
i4-inch  gun  14.52  inches;  which  would  not  only  secure  a  good 
alignment  of  the  grains  in  the  containing  bag,  but  would  also 
give  a  much  less  initial  surface  of  combustion  to  the  charge  and 
would  thus  reduce  the  maximum  pressure. 

The  general  expression  for  the  surface  of  combustion  of  an 
m.p.  grain  with  7  perforations,  in  terms  of  the  thickness  of  web 
burned,  is  by  (i), 

5  =  S0  +  4  *  (3  ™  ~  2  (R  +  7  '))  I  ~  36  n  P 
Differentiating  twice,  we  have 

=  4  TT  (3  m  -  2  (R  +  7  r))  -  72  nl 


There  is,  therefore,  a  maximum  value  of  S  which  occurs  when 

3  m  -  2  (R  +  7  r) 


I  = 


18 


*  Journal  U.  S.  Artillery,  July-August,  1908,  page  9. 


INTERIOR  BALLISTICS 


and  the  maximum  surface  of  combustion  is 

7r(3w-  2(# 


From  these  formulas  are  easily  deduced  the  following: 

1.  When  3  m  —  2  (R  +  7  r)  =  o,  S  is  a  decreasing  function 
of  /  during  the  entire  burning  of  the  web  thickness. 

2.  When  3  m  —  2  (R  +  7  r)  is    equal    to,   or   greater  than, 
iBl0  the  grain  burns  with  an  increasing  surface. 

3.  When  3  m  —  2  (R  +  7  r)  lies  between  o  and  18  10  the  sur- 
face of  combustion  is  at  first  increasing  and  then  decreasing. 

Expression  for  Weight  of  Charge  Burned.  —  If  we  assume 
that  the  entire  charge  is  ignited  at  the  same  instant,  which  is 
practically  the  case  with  an  igniter  at  both  ends  of  the  cartridge, 
the  combustion  of  the  charge  will  be  expressed  by  the  same 
function  that  applies  to  a  single  grain.  Therefore  if  y  is  the 
weight  of  the  charge  burned  at  any  period  of  the  combustion 
and  o>  the  weight  of  the  entire  charge,  we  may  assume  the 
equality  (since  the  weights  are  proportional  to  the  volumes) 


In  this  equation  a  is  always   positive  from  its  definition, 

S  I 
viz.:  a  =  ~TF^-     It  varies  in  value  from  3  (spheres  and  cubes) 

*  o 

to  less  than  unity  (service  multiperf orated  grains).  The 
smaller  OL  is,  cczteris  paribus,  the  less  will  be  the  maximum 
pressure  for  a  given  charge.  Of  the  other  characteristics,  X 
and  fij  either  may  be  positive,  negative,  or  zero,  but  not  both  at 
the  same  time. 

Expressions  for  Initial  Volume  and  Surface  of  Combustion 
of  a  Charge  of  Powder. — Let  N  be  the  number  of  grains  in 
unit  weight  of  powder,  V  the  volume  of  unit  weight  of  water, 
and  §  the  specific  gravity  of  the  powder.  Then,  from  the 
definition  of  specific  gravity, 


COMBUSTION    UNDER    CONSTANT    PRESSURE  73 

V 


NV 


' (23) 


since  we  may  assume  that  the  weights  are  proportional  to  the 
volumes.  The  number  of  grains  in  unit  weight  of  powder  can 
be  counted,  and,  with  the  carefully  moulded  grains  now  in  use, 
V0  can  be  calculated  with  great  accuracy.  Thus  (23)  can  be 
employed  to  determine  the  specific  gravity  of  a  powder  when  it 
is  not  given  by  the  manufacturer,  as  is  usually  the  case.  For 
the  large  grains  designed  for  seacoast  guns  the  number  of  grains 
in  100  units  should  be  counted,  estimating  the  fraction  of  a 
grain  in  excess.  For  small-arms  powder,  if  the  specific  gravity 
of  the  mass  of  which  the  grains  are  made  is  known,  the  number 
of  grains  in  unit  weight  may  be  computed  by  the  formula 

V 

N     =    Jy-^^ (24) 

The  units  to  be  used  in  these  and  other  formulas  that  will 
be  deduced  will  be  considered  later. 

Initial  Surface  of  Unit  Weight  of  Powder  and  of  the  Entire 
Charge. — Let  S±  be  the  initial  surface  of  the  grains  of  unit  weight 
of  powder.  Then  if  S0  is  the  surface  of  one  grain,  we  have,  by 

(24) 

Si  =  NS.=  j^.     .....     (25) 

But  by  (8) 


^o  —        / 
lo 

Therefore 


d  10  10 

for  one  unit  weight  of  powder;  and  for  w  units  weight, 


74  INTERIOR   BALLISTICS 

This  simple  formula  was  first  published  in  the  Journal  U. 
S.  Artillery  for  November-December,  1905.  It  shows  that  for 
two  charges  of  equal  weight  and  made  up  of  grains  of  the  same 
density  and  thickness  of  web,  but  of  dissimilar  forms,  the  entire 
surfaces  of  all  the  grains  in  the  two  charges  are  proportional  to 
the  corresponding  values  of  «.  It  also  shows  that  if  the  initial 
surfaces  of  two  charges  of  equal  weight  but  made  up  of  grains  of 
dissimilar  forms,  are  to  be  the  same,  the  web  thicknesses  must  be 
inversely  as  the  values  of  <*.  For  example,  if  the  two  charges 
are  made  up,  the  one  of  cubes  and  the  other  of  long  slender 
cylinders  (axite  and  cordite),  the  web  thickness  of  the  former 
must  be  one-half  greater  than  the  latter  to  obtain  the  same 
initial  surface  for  each  charge.  These  principles  are  important 
since  the  maximum  pressure  in  a  gun  varies  very  nearly  with  the 
initial  surface  of  the  charge. 

Volume  of  Entire  Charge. — Let  Vs.  be  the  volume  of  a 
charge  of  o>  units  weight  supposed  to  be  reduced  to  a  single 
homogeneous  grain.  For  a  single  grain  of  unit  weight  (23)  gives 

V  -  — 
d 

and  for  d>  units 

*VF •  ^ 

Gravimetric  Density. — Gravimetric  density  is  the  density 
of  a  charge  of  powder  when  the  spaces  between  the  grains  are 
considered.  It  is  measured  by  the  ratio  of  the  weight  of  any  given 
volume  of  the  powder  grains  to  the  weight  of  the  same  volume 
of  water.  Since  one  pound  of  water  fills  27.68  cubic  inches  we  may 
say  that  the  gravimetric  density  of  a  powder  is  the  weight  in 
pounds  of  27.68  cubic  inches  of  the  powder  not  pressed  together 
except  by  its  own  weight.  Or,  if  we  take  a  cubic  foot  as  the 
unit  and  designate  the  gravimetric  density  by  ?%  the  weight  of 
a  cubic  foot  of  the  powder  grains  by  «',  and  by  w  the  weight  of 


COMBUSTION    UNDER    CONSTANT    PRESSURE  75 

a  cubic  foot  of  water,  we  shall  have  by  definition, 


w       1728/27.68       62.427* 

It  is  evident  that  f  will  vary  not  only  with  the  density  of 
the  individual  grains  but  also  with  the  volume  of  the  interstices 
between  them;  and  this  latter  varies  with  the  general  form  of 
the  grains,  or,  in  other  words,  with  their  ability  to  pack  closely 
or  the  reverse.  It  is  evident  that  the  maximum  value  of  f  is 
the  weight  of  a  cubic  foot  of  solid  powder,  in  which  case  the  above 
ratio  would  be  the  specific  gravity  of  the  powder,  designated  by 
8.  The  gravimetric  density  is  therefore  always  less  than  the 
specific  gravity.  For  modern  powders  gravimetric  density  is 
of  very  little  importance. 

Density  of  Loading.  —  Density  of  loading  is  defined  to  be 
the  "ratio  of  the  weight  of  charge  to  the  weight  of  a  volume  of 
water  just  sufficient  to  fill  the  powder  chamber."  Let  A  be  the 
density  of  loading  and  Vc  the  volume  of  the  powder  chamber. 
Since  V  is  the  volume  of  unit  weight  of  water  it  is  evident  that 
Vc/  V  is  the  weight  of  a  volume  of  water  equal  to  the  volume  of 
the  chamber.  Hence  by  definition, 


From  (27)  we  have, 


V 
and  this  substituted  in  (28) ,  gives 


^>  / 

A  =  ~y~ (29) 

c 

From  this  last  equation  the  density  of  loading  may  be  defined 
as  the  ratio  of  the  volume  of  the  powder  grains  supposed  to  be 
reduced  to  a  single  grain,  to  the  volume  of  the  chamber,  multi- 
plied by  the  density  of  the  powder.  If  V^  =  VCJ  that  is,  if  the 


76  INTERIOR   BALLISTICS 

chamber  is  filled  by  a  single  grain,  then  A  =  d  ;  and  this  is  the 
superior  limit  of  density  of  loading.  The  inferior  limit  is,  of 
course,  zero,  namely,  when  V&  =  o.  If  the  density  of  loading 
is  unity  it  follows  from  (28),  that 

Vc 
=  F7' 

that  is,  the  weight  of  charge  equals  the  weight  of  water  that 
would  fill  the  chamber. 

Reduced  Length  of  Initial  Air  Space. — By  initial  air  space 
is  meant  that  portion  of  the  volume  of  the  chamber  not  occupied 
by  the  powder  grains  constituting  the  charge.  The  reduced 
length  of  the. initial  air  space  is  the  length  of  a  cylinder  whose 
cross-section  is  the  same  as  that  of  the  bore,  and  whose  volume 
is  equal  to  the  initial  air  space.  Denote  this  length  by  z0  and 
the  area  of  cross-section  of  the  bore  by  &>.  Then  as  Vc  —  V^ 
is  the  volume  of  the  air  space  we  have 


Substituting  for  Vc  and  V&  their  values  from  (28)  and  (27), 
we  have 

Zo=~VvA~ 

Put 

i        i       d  -A 


A       0          Ad 
Then 


(30) 


Working  Formulas  for   English   and   French  Units. — The 

English  units  used  with  formulas  (23)  to  (30),  inclusive,  are  the 
pound  and  inch.     Therefore 

V  =  26.78  cubic  inches,  nearly. 


COMBUSTION    UNDER    CONSTANT    PRESSURE  77 

The  French  units  employed  with  the  same  formulas  are  the 
kilogramme  and  decimetre.     For  these  units  we  have 

Vf  =  i  cubic  decimetre. 
The  two  sets  of  formulas  in  working  form  are  therefore: —    « 


d 

N 

V, 
A 

•7. 

English 

27.68 

Units 

(230 

(240 

a  *      (->fr'\ 

French 

*          J 

Units 

(23") 
(24") 
(26") 

(27") 
(28") 
IW» 

"  NV0 
27.68 

~NV0 

N         ' 

"  dV0 

27.68 

^   w. 

a  co 

27.68* 

(20  ) 
•> 

"             <W^ 

F    -  - 

o- 

27.68 

\27  ) 
&         (<?%'} 

V»       d 

CO 

A  =  V 

9.       —    

vc 

27.68 

(2Q  ) 
-     *        (***\ 

EXAMPLES 

i.  Compute  the  number  of  grains  in  a  pound  of  the 
powder  used  with  the  service  magazine  rifle.  Also  the  initial 
surface. 

The  grains  of  this  powder  are  pierced  cylinders  of  the  follow- 
ing dimensions: 

R  =  o.".o45 
r  =  0.015       .'.     2  10  =  o/'.o3 

i 

m  =  -  -  in. 
21 

*  =  1.65 

OL    =    1.63 

CO    =    I 


78  INTERIOR  BALLISTICS 

From  (24')  and  (26'),  we  have 
27.68 

N  =  --  ;  -  75  --  rr-T  =  62300 
4  TT  10  m  (R  -  10)  d 

27.68  X  1.63 

Si  =  -~-    --  -  =  1823  in.2 
1.65  X  0.015 

It  is  officially  stated  that  the  number  of  grains  per  pound 
varies  from  83,000  to  93,000.  This  discrepancy  is  partly  due  to 
shrinkage  and  partly  to  the  breaking  and  chipping  of  the  grains. 
Possibly  also  to  the  method  of  counting. 

2.  What  is  the  entire  initial  surface  of  a  charge  of  70  Ibs.  of 
the  m.p.  powder  designed  for  the  8-inch  rifle  ?  For  this  powder 
we  have 


R  =  o".256;  r  =  o.".o255;  m  =  i".O29;  d  =  1.58 
-  -  =  o".044875; 
27.68  X  0.72667  X  70 


10=  -  -  =  o".044875;  «  =  0.72667 


•  •     ,  =  19813  in." 

1.58  X  0.044875 

3.  Suppose  the  powder  of  example  2  to  be  made  into  cubes 
having  the  same  thickness  of  web.  What  would  be  the  initial 
surface  of  the  charge? 

For  a  cube  OL  =  3.     Therefore 


To  make  the  initial  surface  of  the  latter  charge  the  same  as 
the  former  the  web  thickness  would  have  to  be 

3  X  0.08975         „ 
2  1°         0.72667  >37 

4.  The  volume  of  the  chamber  of  the  1  2-inch  rifle  is  17487 
cubic  inches.  If  the  charge  is  400  Ibs.  what  is  the  density  of 
loading?  Ans.:  A  =  0.633. 


CHAPTER    IV 

COMBUSTION    AND    WORK    OF   A    CHARGE    OF    POWDER   IN 

A    GUN 

IT  has  been  established  by  experiment  that  a  grain  of  modern 
powder  burns  in  concentric,  parallel  layers,  and  that  the  velocity 
of  combustion  under  constant  pressure  is  uniform.  Let  10  be 
one-half  the  web  thickness  of  a  grain  and  r  the  time  of  burning 
this  thickness  under  the  constant  pressure  of  the  atmosphere. 
We  then  have,  since  the  web  burns  on  both  sides, 

7  =  velocity  of  combustion  =  constant  =  vc  (say) .       (i) 

In  the  bore  of  a  gun,  however,  the  pressure  surrounding  the 
grain  is  very  far  from  being  constant  and  greatly  exceeds  the 
atmospheric  pressure.  All  writers  on  interior  ballistics  agree 
that  the  velocity  of  combustion  may  be  regarded  at  each  instant 
as  proportional  to  some  power  of  the  pressure;  but  they  differ 
widely  among  themselves  as  to  what  this  power  is.  Sainte- 
Robert,  Vieille,  Gossot,  and  Liouville  give  reasons  (based,  how- 
ever, upon  experiments  made  with  a  small  quantity  of  powder 
exploded  in  an  eprouvette  of  a  few  cubic  inches  capacity)  for 

2  Q 

adopting  the  exponent  -.     Centervall  makes  the   exponent  — 

for  "Nobel  N  K"  powder.  Sebert  and  Hugoniot,  from  ob- 
servations of  the  recoil  of  a  lo-cm.  gun  mounted  on  a  free-recoil 
carriage,  deduced  a  law  of  burning  directly  proportional  to  the 
pressure.  This  law  is  the  most  simple  of  all  and  allows  an  easy 
and  complete  integration  of  the  equations  entering  into  the 
problem.*  But  simple  as  is  this  law  of  Sebert  and  Hugoniot, 

*  See  Journal  U.  S.  Artillery,  vol.  7,  pp.  62-82. 
79 


80  INTERIOR   BALLISTICS 

we  prefer  to  make  use  of  Sarrau's  law  of  the  square  root  of  the 
pressure,  because  the  resulting  formulas  are  easily  worked 
and  give  results  which  "agree  very  well  with  facts"  as  stated  by 
Sarrau,  and  as  has  been  repeatedly  shown  by  the  writer  and 
others. 

Sarrau's  law  of  burning  under  a  variable  pressure  p  leads 
directly  to  the  equation, 

dl      10P\"  ... 


in  which  p0  is  the  atmospheric  pressure  and  /  the  thickness  of 
layer  burned  in  time  t. 

It  will  be  assumed  that  the  variable  pressure  p  in  the  bore 
is  measured  by  the  energy  of  translation  imparted  to  the 
projectile  (which  is  many  times  the  sum  of  all  the  other  energies 
entering  into  the  problem) ;  and  it  will  be  taken  for  granted  that 
all  the  other  work  done  by  the  expansion  of  the  powder  gas  may 
be  accounted  for  by  giving  suitable  values  to  the  constants  so 
as  to  satisfy  the  firing  data  by  means  of  which  they  are  deter- 
mined. This  procedure  will  be  fully  illustrated  further  on. 

If  p  is  the  variable  pressure  per  unit  of  surface  upon  the  base 
of  the  projectile  at  any  instant,  &  the  area  of  the  base,  and  u 
the  corresponding  distance  travelled  by  the  projectile  from  its 
firing  seat,  we  have  from  the  principle  of  energy  and  work, 

in  which  w  is  the  weight  of  the  projectile. 
But  from  mechanics  and  calculus, 

d2u      dv      d  v      du         dv       id  (v2) 

dt2  ~  dt  ~  du      dt        Vdu  "~  2    du  ' 

in  which  v  from  now  on  represents  velocity. 
Therefore 

w    d(*) 


COMBUSTION     OF     A     CHARGE     OF     POWDER     IN     A     GUN       8l 

Combining  (2)  and  (3),  we  have 

*!  =  !*.(—  w--\*    (<LW1\* 

dt      ~~     T   \2gupJ          \     du    )  •         •         •        U) 

Since  z0  is  the  reduced  length  of  the  initial  air-space  and  u 
the  distance  travelled  by  the  projectile  from  its  firing  seat,  we 
may  say  very  approximately,  by  the  principle  of  the  covolume, 
that  u/z0  is  the  number  of  volumes  of  expansion  of  the  gas  during 
the  travel  u,  and  this  whether  the  charge  is  all  converted  into 
gas  or  not.  If  we  make  u/z0  =  x,  and  therefore  du/d  x  =  z0, 
(3)  and  (4)  become 

w      d 


and 


co  p  =  --  •  —  r  —        .....      (c) 
2  g  Z0       dx 


dt    '  r\2     uz        \  dx 


It  will  be  seen  that  (6)  connects  the  velocity  of  burning  of 
the  grain  with  the  velocity  of  travel  of  the  projectile  in  the 
bore.  It  will  be  better  to  make  x  the  independent  variable  in 
the  first  member  as  well  as  the  second. 

We  have  from  calculus, 

dl       dl    d  x    du       v     dl 
dt      dx   du    dt  ~  z0    dx 

Therefore,  substituting  in  (6), 

wz 


dx       r  \2  gu>  po'     \  d  x        v 
Integrating  between  the  limits  o  and  x,  we  have 


In  order  to  perform  the  integration  indicated,  we  must  know 
the  relation  existing  between  v  and  x,  that  is  between  the  velocity 

of  the  projectile  in  the  bore  at  any  instant  and  the  corresponding 
6 


82  INTERIOR   BALLISTICS 

number  of  volumes  of  expansion  of  the  gas.     We  get  this  relation 
from  (19)  Chapter  II,  which  is 


w 
From  this  equation  we  deduce  by  simple  differentiation 


dx      v 
Substituting  this  in  (7)  and  making 

r*  d  x 

x>  =J  voT^FTcrr 

we  have 


It  will  be  observed  that  X0  is  a  function  of  a  ratio  and  is  in- 
dependent of  any  unit,  and  may  therefore  be  tabulated  with  x 
as  the  argument. 

If  we  put 


0 
K  —  —  I-  ---  —  )       .....     (n) 

r  V6£«#a/ 

we  have 

j  =  KX.      ......     (12) 

10 

Substituting  the  value  of  Ill0  from  (12)  in  (22),  Chapter  III, 
we  have 

k  =-1  =  aKX0(i  +  \KX0+  v(KXoy-)      .      (13) 

CO 

an  equation  which  gives  the  fraction  of  the  charge  burned  at  any 
instant  in  terms  of  the  volumes  of  expansion  of  the  gases  gener- 
ated. When  the  powder  is  all  burned  in  the  gun  (if  it  be  all  burned 
before  the  projectile  leaves  the  bore),  we  have  y  =  co  and  I  =  10. 


COMBUSTION    OF    A    CHARGE    OF    POWDER    IN    A    GUN       83 

If,  therefore,  we  distinguish  X0  by  a  dash  when  /  =  10)  (12) 
becomes 

KX0  =  i,  ......     (14) 

and  (13)  reduces  to 

i  =  a(i  +  X  +  /0 

a  fundamental  relation  established  in  Chapter  III. 

Substituting  the  value  of  K  from  (14)  in  (13),  we  have,  while 
the  powder  is  burning,  the  relation 

X  X 


Expression  for  Velocity  of  Projectile  while  the  Powder  is 
Burning.  —  Substituting  the  value  of  y  from  (15)  in  (8)  and 
making 

*'=*•('-  (FT^ji)     •    •    •    •    ^ 

we  have 

»  co       X  j  (  X0  fXo\  2\ 

tf  =6gaf-  '=-Ji+X  =-  +  M  =  )    \.      - 
w    Xo  (  X0  \A0/    ) 

This  equation  holds  only  while  the  powder  is  burning  and 
ceases  to  be  true  when  X0  >  X0. 

Velocity  of  Projectile  when  y  =  co.  When  X0=  X0  and, 
therefore,  Xl  =  Xi,  equation  (17)  reduces  to 

V  =  6g/-=r<*(i  +  X  +  //); 
w  X2 

or,  since 

"  (i   +  X  +  /£)  =  i, 
it  becomes 


This  equation  is,  of  course,  the  same  as  (8)  from  which  it  is 


84  INTERIOR  BALLISTICS 

derived  as  is  evident  from  (16).     Putting 

A"t  _- 

^  ~  X2> 
or,  generally, 


the  expression  for  v  2  becomes 


It  should  be  remembered  that  all  symbols  employed  in  this 
work  affected  with  a  dash  refer  to  the  position  of  the  projectile, 
either  in  the  bore  on  in  the  bore  prolonged,  when  the  powder 
has  all  been  burned,  and  therefore  where  y  =  o>. 

From  (19),  we  have 

,     ,       v2w 

6  gf  =  =•  r  ; (20) 

and  this  substituted  in  (17),  gives 

.^«vMi+xfs+>Y|fyt         («) 


For  convenience,  put 

•A-l 

Then,  finally,  while  the  powder  is  burning, 


•A-l  J*-  **•  ^ 


tf=MXi{i+NX0+N'X0*}    .     .     .     (22) 

Velocity  of  Projectile  after  Powder  is  all  Burned.—  The 

velocity  of  the  projectile  after  the  powder  is  all  burned  is  given 
by  (8),  substituting  &  for  y.  Reducing  by  means  of  (16),  (18), 
and  (20),  and  denoting  velocity  after  the  powder  is  all  burned 
by  capital  F,  equation  (8)  becomes 


COMBUSTION    OF    A    CHARGE     OF    POWDER    IN     A    GUN       85 

The  velocity  then  after  the  powder  is  all  burned  varies 
directly  as  the  square  root  of  X2.     From  (16)  and  (18),  we  have 


and  therefore  the  superior  limit  as  x  (or  «)  increases  indefi- 
nitely is  unity.     On  this  supposition  (23)  becomes 

r-  =  =  =  V?  (say)        ....     (25) 

A2 

We  may  regard  Vi  then  as  the  theoretical  limiting  velocity 
after  an  infinite  travel.     In  terms  of  V\  (23)  becomes 

F2=712JT3     ......     (26) 

Since  from  (25)  v2  =  X2  V\  !,  therefore 


and,  therefore, 

,,       «Ft2 

^  :    -=— (27) 

Pressure  on  Base  of  Projectile  while  Powder  is  Burning. — 

Differentiating  (17)  with  respect  to  the  independent  variable  x 
and  putting  for  simplicity 


we  have 


Therefore,  from  (5) 

W 


Combining   the   constants   outside    the   brackets   into   one 
multiplier  by  making 

-  -  M', 


86  INTERIOR   BALLISTICS 

we  have  the  following  expression  for  the  pressure  per   unit  of 
surface,  on  the  base  of  the  projectile : — 

Pressure  after  the  Powder  is  all  Burned. — Differentiating 
(26)  with  reference  to  x  and  substituting  the  differential  co- 
efficient in  (5)  we  have,  employing  capital  P  to  express  pressure 
in  this  case, 

wV*    dX2 
2  g co z0  dx 
But  from  (24) 


dx       3 

(i  +x)*' 

wV,2 

p' 

do) 

6  g  oj  2 
P 

P' 

\^u/ 

(*T\ 

whence,  putting 
we  have  finally 

If  we  make  x  =  o  in  (31),  we  have 

p  =  f. (32) 

Therefore  P'  is  the  pressure  per  unit  of  surface  at  the  origin 
supposing  the  powder  to  be  all  burned  before  the  projectile 
moves  from  its  seat. 

Relation  Between  f  and  P'. — From   (19)  and  (25)  we  get 


Combining  this  with  (30)  there  results 

P'  =—J~  (34) 

Z0  co 

Since  /  is  (at  least  theoretically)  the  pressure  per  unit  of 
surface  of  the  gases  of  one  pound  of  powder  at  temperature  of 
combustion,  occupying  unit  volume,  it  follows  from  (34)  that 


COMBUSTION    OF    A    CHARGE    OF    POWDER    IN    A    GUN         87 

P'  is  the  pressure  per  unit  of  surface  of  the  gases  of  co  pounds 
of  powder  (the  entire  charge),  occupying  a  volume  equal  to  the 
initial  air  space  z0  %  as  has  already  been  shown  by  equation  (32). 
Equation  (31)  is,  therefore,  the  equation  of  the  pressure  curve 
upon  the  supposition  that  the  charge  is  all  converted  into  gas 
before  the  projectile  has  moved  from  its  seat.  From  equation 
(30'),  Chapter  III,  we  have, 

27.68 
z0  u  = r-  a  co    cubic  feet. 

Therefore,  from  (34) 


27.68  a 

Values  of  the  X  Functions. — These  values  may  be  most 
easily  and  simply  expressed  by  means  of  auxiliary  circular 
functions.  Thus  let 

(i  -f  x)*  =  sec0 (36) 

Then,  by  trigonometry, 

sin2  0  =  i— TT  =  X2.     (from  (22)) 

and 

tan  0  =  V(i  -f-  #)i  —  i 
Also 

d  x  =  6  sec6  0  tan  0  d  0 

Substituting  these  values  in  the  expression  for  X0  we  have 

X0=  6Jo  sec30</0 (37) 

Integrating,  we  have 

X0=  3  sec  0  tan0  +  3  loge  (sec  0  +  tan0)        .     (38) 

By  substituting  the  values  of  sec  0  and  tan  0  given  above  in 
(38),  we  get  an  expression  for  X0  in  terms  of  x.  But  it  is  of  no 
practical  interest.  The  definite  integral  in  (37)  is  a  well-known 
function  of  0  and  has  been  extensively  used  in  exterior  ballistics. 


88  INTERIOR   BALLISTICS 

A  table  of  this  function  computed  for  every  minute  of  arc  up  to 
87°  was  first  published  by  Euler,  and.  has  recently  been  re- 
printed (1904)  at  the  Government  Printing  Office  and  issued 
as  "Supplement  No.  2,  to  Artillery  Circular  M."  By  means  of 
this  table  it  is  easy  to  compute  X0  for  any  value  of  x.  First 
compute  0  by  (36),  and  then  take  the  definite  integral  corre- 
sponding to  0,  which  has  been  symbolized  by  (0),  from  the  table 
just  mentioned.  We  then  have 

X0  =  6  (0)     ......     (39) 

Since 

X2  =  sin20  .......     (40) 

we  have  from  (18) 

X^X^m't        .....     (41) 

By  definition 

x      dXl 
x>- 


But  from  (41) 


dX 


sin*  cos* 


From  (9)  and  (36)  we  deduce 


dX0   . 

-7 —  sm  0  =  sin  0  cos  0. 

d  x 

Also,  from  what  precedes, 

d  0        sin  0  cos  0        cos8  0 

2  sin  0  cos  0  -j—  = ^ = . 

djc     3sec80tan0         3 

Therefore 

i 
A  3  =  sin  0  cos  0  +  —X0  cos  0. 

o 

Let 

X  = - 

i  +  i  X0  cos4  0  cosec  0 


COMBUSTION    OF    A    CHARGE    OF    POWDER    IN    A    GUN       89 

Then  we  have    'v/cuiQ  , 

sin  0  cos4  0 
Xa  = Y~          (^ 

From  the  foregoing  equations  we  find 

X3  dx 

by  means  of  which  are  easily  deduced  from  the  definitions  of 
X4  and  X*,  the  following  simple  equations: — 

X*  =  X0  (i  +  X) (43) 

and 

V  V  2   (          I      ~    V\  f  .  .\ 

X^=  X02  (i  +  2  A)        ....      (44) 

By  means  of  equations  (39),  (40),  (41),  (42),  (43)  and  (44), 
the  table  of  the  logarithms  of  the  X-functions  given  at  the  end 
of  the  volume  was  computed. 

Some  Special  Formulas.  Dividing  (21)  by  (15)  and  reducing 
by  (25),  we  have,  since  yl&=  k, 

7.2  _  z,  yi  _  z,  y  2  v-  /    \ 

V       -   K    V        -    K    V  i  vV2  .         .         .         .         V45/ 

That  is,  the  velocity  of  the  projectile  at  any  travel  before  the 
charge  is  all  burned  is  equal  to  what  the  velocity  would  have 
been  at  the  same  travel  had  all  the  charge  been  converted  into 
gas  before  the  projectile  moved,  multiplied  into  the  square  root 
of  the  fraction  of  charge  burned. 

For  spherical,  cubical,  and  certain  other  forms  of  grain,  we 

have  a  =  3,   \  =  —  i  and  n  =  — .     Substituting  these  in  (15), 

o 
we  have  by  obvious  reductions, 

k  =  i  —  (  i  —  -=2  j (46) 

and  therefore 

x0  =  x0{  i  -(i  -*)M    ....    (47) 


QO  INTERIOR   BALLISTICS 

For  cordite  and  similar  grains  we  have  a=  2,  X  =  -    -  and  ^ 
=  o.     Substituting  these  in  (15),  gives 

*-'-('-=;)'    ....  (48) 

and 

X0=X0{  i-(i  -*)»}  ....  (49) 
Equations  (46)  and  (48)  give  the  fraction  of  the  charge  consumed 
for  any  given  travel  of  the  projectile,  and,  conversely,  (47)  and 
(49)  enable  us  to  determine  the  travel  of  projectile  for  any  given 
fraction  of  charge  burned.  For  any  other  forms  of  grain  the 
solution  of  a  complete  cubic  equation  is  necessary  to  determine 
X0  when  k  is  given.  See  equation  (15). 

Expressions  for  Maximum  Pressure. — It  is  well  known 
that  the  maximum  pressure  in  a  gun  occurs  when  the  projectile 
has  moved  but  a  comparatively  short  distance  from  its  seat,  or 
when  u  and  x  are  relatively  small.  The  position  of  maximum 
pressure  is  not  fixed  but  varies  with  the  resistance  encountered. 
As  a  rule  it  will  be  found  that  the  less  the  resistance  to  be  over- 
come by  the  expanding  gases  the  sooner  will  they  exert  their 
maximum  pressure,  and  the  less  will  the  maximum  pressure  be. 
The  differentiation  of  (29)  gives  an  analytical  expression  for 
the  maximum  value  of  p;  but  it  is  too  complicated  to  be  of  any 
practical  use.  A  reference  to  the  table  of  the  X  functions  shows 
that  Xz  is  approximately  a  maximum  when  x  =  0.64,  while  X± 
and  X5  increase  indefinitely.  When  X  is  negative  it  is  evident 
that  p  is  a  maximum  when  x  is  less  than  0.64;  and  when  X  is 
positive,  when  x  is  greater  than  0.64.  Therefore  there  will  be 
two  cases  depending  upon  whether  the  grains  burn  with  an  in- 
creasing or  a  decreasing  surface.  These  will  be  considered 
separately. 

(a)  When  the  grains  burn  with  a  decreasing  surface;  or  what 
is  the  same  thing,  when  X  is  negative.  A  function  at,  or  near, 
its  maximum  changes  its  value  slowly.  Therefore  a  moderate 


COMBUSTION     OF     A    CHARGE     OF     POWDER     IX     A    GUN       91 

variation  of  the  position  of  maximum  pressure  will  have  no 
practical  effect  upon  its  computed  value.  It  has  been  found 
by  trial  in  numerous  cases  that  x  =  0.45  gives  the  position  of 
maximum  pressure  when  X  is  negative  with  great  precision. 
For  this  value  of  x  the  table  gives, 

log  X3  =  9.85640  —  10 

log  X4  =  0.48444. 

logXb  =  0.93587. 

Substituting  these  in  (29)  and  designating  the  maximum 
pressure  by  pml  we  have  approximately,  when  X  is  negative, 

pm  =  [9.85640  -io]M'{i-  [0.48444]  ^V  +  [0.93587]  N'  }   (50) 

or,  , 

pm=  0.71846  M'{i  -  3.0510^  +  8.6273^']     .   (50') 

(b)  When  the  grains  burn  with  an  increasing  surface.  When 
the  grains  burn  with  an  increasing  surface  X  is  generally  positive, 
and  it  will  not  be  far  wrong  to  assume  that  the  maximum  pressure 
occurs  when  x  =  0.8.  For  this  value  of  x  the  table  gives: 

log  X3  =  9.86027-10. 
log  X±  =  0.60479 


Substituting  these  in  (29),  we  have, 

pm  =  [9.86027  -io]M'{  i  +  [0.60479]  #-[1.17352]  #')    (51) 
or, 

pm=  0.72489^(1  +4-0252  N-  14.911  #'}   .     (51') 

Expressions  for  Computing  r  and  the  Velocity  of  Combustion. 
From  (n)  and  (14)  we  have 

(    ™    *o     \*=  ,       . 

:=l<^t/*°  .....  (52) 

If  vc  is  the  velocity  of  combustion  under  atmospheric  press- 
ure we  shall  have 

/, 

*<=-•> 


92  INTERIOR   BALLISTICS 

and  therefore 

(6g<*Po\*lo  ,      . 

V<=-(-^Z-)YO    •  -  -  -  (53) 

Let  v'c  be  the  velocity  of  combustion  at  any  instant  under 
the  varying  pressure  p.     Then  from  (2)  we  have 


Working  Formulas.  English  Units.  —  It  is  customary  in 
our  service,  following  the  English  practice,  to  express  the  volumes 
of  the  powder  chamber  and  bore  in  cubic  inches;  the  various 
pressures  in  pounds  per  square  inch;  the  caliber,  reduced 
length  of  initial  air  space,  and  travel  of  the  projectile  in  the 
bore,  in  inches;  while  the  velocity  of  the  projectile  is  expressed 
in  foot-seconds  and  its  weight  in  pounds  and  ounces.  These 
units  are  apt  to  cause  confusion  and  error  in  the  applications  of 
ballistic  formulas;  and  to  avoid  this  as  much  as  possible  it  will 
be  well  to  reproduce  the  most  important  of  the  formulas  deduced 
in  the  preceding  pages  with  all  the  reductions  made  and  the 
mathematical  and  physical  constants  introduced  and  combined 
into  one  numerical  coefficient.  The  physical  constants  adopted 
for  English  units  (foot-pound),  are  the  following: 

g  =  32.16  f.s.  (mean  for  the  United  States) 
p0=  14.6967  Ibs.  per  in.2 
V  =  27.68  cubic  inches. 

/  is  taken  in  pounds  per  square  inch.     The  formulas  are  re- 
numbered for  convenience. 

A  =  27.68^-  =  [1.44217]^-      ......  (54) 

'  c  v  c 

I         i        8  -  A 

a  =  T~i  =  TF      ......    •    (S5) 

4X27.68    a  Si       r  oi'7"/-     u    \  /  *\ 

Z0=  -  ~  •  -^  =  I*  -54708]  js-  (inches)       .      (56) 


COMBUSTION     OF     A      CHARGE     OF  POWDER     IN    A    GUN  Q3 

w-OLCl 

(57) 


Ff  =  144  X  6  gf-^  =  [4.44383]^  (foot-seconds)     .      .      (58) 

_r_       V*  =  MX0  =  \M  =  sMP'  _v> 

kX2       X2         a         aN  '       Mf         X2 

av-      a  V,2 

M  =  -rr=-=-   ............     (    ) 

AI          A0 

6        iv  M      r  ,  w  M 

M'=  -  -=   7.82867  -  10  -          .      .  .  .      .      .     (61) 

27.68^     as,  as, 


aP'       1728 

T  =  ^68  .............     (65) 

X  27.68     X0Vaws,     r  X0  Va  w  a    ,,  , 

'          ^      =[8-56006-10]  ---    - 


7  J    /72 

vc  =  -  =  [i.43994  ]  -^    /—-  =  (inches  per  second)     .      .     (67) 
r  X0Va  w  a, 

ve'  =  vc          2  =  [941639  -  10]  vc  V7  ....     (68) 


v        fi  -43994]  r^3       [1.43994]  lo  d2 

XQ=  -  --  -j==--   .......     (69) 

V  aw  w  vc\/  aw  & 


It  must  be  remembered  that  v  and  />  refer  to  the  period  when 
the  powder  is  burning  and  V  and  P  to  the  period  after  the  powder 
is  all  burned. 


94  INTERIOR  BALLISTICS 

FRENCH  UNITS 

In  metric  units  we  shall  take  Vc  in  cubic  decimetres,  p  in 
kilogrammes  per  square  centimetre,  d  in  centimetres  and  z0,  u 
and  v  in  metres.  Also  g  =  9.80896  m.s.  With  these  units-  our 
formulas  become, 

A  =  £       - (71) 

40    a  co      r              ,  a  s, 
Z0  =  —  •  -jT  =  11.10491]  -# (72) 

f  ~ 
Vf  =  [2.76977]  -^  (f  in  kilos,  per  cm.2)  .       .      .     (73) 

w  M 
M'=  L7.70735  -  10]  —  T  (tilos.  per  cm.2)     .      .     (74) 

d     CO 

wV* 

P' '=  [7.23023  —  10]  —     -  (kilos,  per  cm.2)    .      .     (75) 
a  co 

wV2 
f  =  [7.23023  —  10]  -      -  (kilos,  per  cm.2)    .      .     (76) 

CO 

vc  =  [0.63128]         ° (cm.  per.  sec.)       .      .    (77) 

X0  V  a  w  co 

/  d2  T  d2 

X0=  [0.63128] — —  =  [0.631-28]—=.  .     (78) 

V  aw  & 


T    =    [9.36872    -    10] -^  •         (79) 

Characteristics  of  a  Powder. — The  quantities  /,  T,  a,  X 
and  M  were  called  by  Sarrau  the  characteristics  of  the  powder  be- 
cause they  determine  its  physical  qualities.  Of  these  quan- 
tities /  depends  principally  upon  the  composition  of  the  powder, 
and,  with  the  same  gun,  for  service  charges,  is  practically  constant 
for  all  powders  having  the  same  temperature  of  combustion. 
The  value  of  r  depends  generally  upon  the  density  and  least 


COMBUSTION     OF     A    CHARGE     OF     POWDER     IN     A     GUN       95 

dimension  of  the  grain.  The  factors  a,  X  and  (JL  called  "form 
characteristics"  depend  upon  the  form  of  the  grain,  and  for 
the  carefully  moulded  powders  now  employed  their  values  may 
be  determined  with  great  precision.  They  are  constant  so  long 
as  the  grain  in  burning  retains  its  original  form. 

Expressions  for  M,  M',  N  and  N'  in  Terms  of  the  Charac- 
teristics of  the  Powder.  —  When  /  and  vc  are  known  from  ex- 
perimental firings  or  otherwise,  for  any  gun  and  powder,  the 
quantities  V?  and  X0  can  be  determined  either  from  (58)  and 
(69),  or  (73)  and  (78).  Substituting  these  in  the  proper  expres- 
sions for  M,  M',  N  and  N'  they  become 

For  English  Units 

....     (80) 


If-  [0.83356]  --          -...     (81) 


N  =  [8.56006-  IO]-KJ  v7 aw n,    .      .      .  (82) 

AT'=£A"  •    • (83) 

For  Metric  Units 

lrfx*    •  •  •  (84) 

-,    <-*.   /     u  c  I    w    w  \  •*• 

tf'=[o.igoo3]-jM--)          .      .      .  (85) 


N  =  [9.71291  -  io]Vawu      .      .      .     (86) 

AT'=  %N* (87) 

If  we  substitute  the  value  of  M'  from  (81)  in  (50)  or  (51),  and 
reject  the  terms  within  the  brackets,  we  have  in  effect  Sarrau's 


96  INTERIOR   BALLISTICS 

monomial  formula  for  maximum  pressure.  But  it  is  evident 
there  can  be  no  monomial  formula  for  velocity  or  pressure  unless 
X  and  fj,  are  approximately  zero.  Equations  (80)  to  (87)  are 
useful  for  determining  the  values  of  M  and  N  (upon  which  all 
the  other  constants  depend),  when  the  charge  varies  or  when 
there  are  variations  in  the  weight  of  the  projectile.  In  these 
formulas  a-,  X,  n  and  10  are  independent  of  &  and  d  and  are  strictly 
grain  constants.  vc  is  a  powder  constant,  varying  only  with 
the  composition  and  density  of  the  powder.  /  is  approximately 
constant  for  full  service  charges  of  the  same  kind  of  powder,  in 
guns  of  all  calibers.  For  example  the  magazine  rifle,  caliber 
0.3  inches,  and  the  1  6-inch  B.  L.  R.  give  approximately  the  same 
value  to/  when  computed  by  equation  (64)  or  (65).  This  factor, 
however,  varies  with  the  charge  in  the  same  gun,  for  it  is  evident 
that  its  effective  value  as  measured  by  projectile  energy  must 
decrease  with  the  charge.  Indeed  if  the  charge  be  sufficiently 
reduced  it  is  obvious  that  /  becomes  zero  since  we  have  omitted 
from  our  formulas  all  consideration  of  the  force  necessary  to 
start  the  projectile.  The  law  of  variation  is  not  known;  but 
we  will  assume  provisionally  that/  varies  with  the  charge  accord- 
ing to  the  law  expressed  by  the  equation 


/-/.•      .....  (88) 

where  co0  is  the  service  charge  by  means  of  which  M  and  N  were 
determined  and  f0  the  corresponding  value  of  /  computed  by 
(64)  or  (76).  If  the  weight  of  the  projectile  also  varies  we  will 
assume  that  /  may  be  determined  by  the  equation 


The  exponents  n  and  nf  must  be  determined  from  experi- 
mental data.     If  we  make 

K  —  -~-r> (90) 

co     W 


COMBUSTION    OF    A    CHARGE    OF    POWDER    IN    A    GUN        97 

(89)  becomes 

/  =  K  «"  wn> (90') 

Substituting  this  expression  for  /  in  (80)   and  (84)   gives,  for 
English  units: 

^b^^-ff^)*.      .      .     (91) 
and  for  metric  units: 


M  =  [2.48268]  K  -^yV^^r^J'     -     •    (91') 

In  the  applications  of  these  equations  f0  must  be  computed 
by  (64)  or  (76)  and  vc  by  (67)  or  (77). 

7 


CHAPTER  V 

APPLICATIONS 

THE  principal  formulas  deduced  in  Chapter  IV  are  here  re- 
produced for  convenience  of  reference.  They  are  the  following: 

(a)  Formulas  which  Apply  Only  While  Powder  is  Burning.— 

v2  =  MX,  [i  +  NX0+N'X02}      .     .     .     (i) 
p  =  M'X3{i+NXt+N'Xs}      ...     (2) 

It  will  be  observed  that  these  formulas  for  velocity  and 
pressure  are  identical  in  form,  and  that  the  constants  within  the 
brackets  are  common  to  both.  Also  that  M'  is  a  simple  multi- 
ple of  M.  Moreover,  from  the  manner  of  deriving  p  from  v2,  the 
velocity  and  pressure  deduced  from  these  formulas  correspond 
at  every  point  so  that  one  can  be  easily  and  exactly  computed 
from  the  other  without  the  necessity  of  laying  down  velocity 
curves  in  order  to  obtain  the  pressures. 

(b)  Formulas  which  Apply  Only  After  the  Powder  has  All 
Been  Burned.— 


...     (3) 
a 

p=(iw  =  p'(<-^    ...  -  (4) 

(c)  Formulas    Which    Apply    at    the    Instant    of    Complete 
Combustion.  — 

?=  MX,{i  +  NX0  +  N'X02},  (from  (i)):  .      .     (5) 

and  _       Mlti  ,  ,. 

v2  =  —  —  (from  (3))        ....      (5  ) 

Equations  (5)  and  (5')  give  the  same  value  to  zr,  since  the 


APPLICATIONS  99 

former  equation  reduces  to  the  latter_at  the  point  w.  But  (i) 
and  (3)  are  not  tangent  at  the  point  w  unless  the  vanishing  sur- 
face (5r)  of  the  grain  is  zero,  as  with  cubes,  spheres,  solid  cylin- 
ders, etc. 

From  (2)  we  have  at  the  travel  « 

P  =  MfX*{  i  +  N  Xt  +  N'X5}      .     .     .     (6) 
and  from  (4) 

'' 


Equations  (6)  and  (6')  give  the  same  value  to  p  for  all  grains 
whose  vanishing  surface  is  zero,  as  may  be  thus  shown  : 

Substituting  for  M'  in  (6)  its  value  from  (59),  Chapter  IV, 
and  giving  to  N  and  N'  their  values  in  terms  of  X0,  we  have 


But        =  i  +  X  and  =^  =  i  +  2  X. 


For  all  grains  whose  vanishing  surface  is  zero  we  have  the 
relations  (Equs.  (10)  and  (12),  Chapter  III.) 

a  (i  +  X  +  M)  =  i 
and 

a  (X  +  2  M)  =  -i 


which  readily  reduces  to  (6').  Therefore  for  all  forms  of  grain 
whose  vanishing  surface  is  zero  the  pressure  curves  (2)  and  (4) 
are  tangent  at  u.  This  is  not  true  for  grains  for  which  5">o. 


100  INTERIOR   BALLISTICS 

For  these  the  pressure  at  travel  u  given  by  (2)  is  greater  than 
that  given  by  (4),  and  this  difference  increases  with  5". 

Monomial  Formulas  for  Velocity  and  Pressure  While  the 
Powder  is  Burning.  —  The  expressions  for  velocity  and  pressure 
while  the  powder  is  burning  (equations  (i)  and  (2))  are  generally 
trinomials  because  equations  (9)  and  (22),  Chapter  III,  are  tri- 
nomials. And  these  are  so  because  of  the  geometrical  character- 
istics a,  X  and  IJL.  In  order  to  have  a  monomial  expression  for 
velocity  or  pressure  X  and  n  must  both  be  zero.  But  upon  this 
supposition  (22),  Chapter  III,  would  become 


or,  the  fraction  of  grain  burned  would  be  directly  proportional 
to  the  thickness  of  layer  burned;  which  is  impossible,  since  the 
grain  burns  on  all  sides.  This  same  supposition  would  also 
make 

I.S.-V, 

which  is  not  true,  at  least  for  finite  volumes. 

It  has  been  shown  in  Chapter  III,  that  all  grains  which,  under 
the  parallel  law  of  burning,  retain  their  original  form  until  wholly 
consumed  and  for  which  5">o,  have  one  or  the  other  of  the 
following  expressions  for  a,  namely,  i  +  x  or  2  +  x,  x  being  the 
ratio  of  the  thickness  of  web  to  the  length  (or  breadth)  of  the 
grain.  Only  grains  for  which  a  =  i  +  x  can  give  approximate 
monomial  expressions  for  velocity  and  pressure,  and  this  by 
making  x  so  small  that  it  may  be  omitted  in  comparison  with 
unity,  in  which  case  a  becomes  practically  unity  and  X  and  /* 
zero.  To  this  class  belong  thin,  flat  grains  and  long  cylindrical 
grains  with  axial  perforation. 

When  a  =  i  and  X  and  /z  are  zero,  equations  (i)  and  (2)  be- 
come 

v2  =  M  X,  .    (7) 


APPLICATIONS  10  1 

and 

p  =  M'XZ  =  [7.82867  -  10]  M—  X3  (8) 

a  & 

Also,  by  equation  (3),  since  a  =  i,  we  have,  after  the  powder 
is  all  burned, 


It  will  be  seen  that  the  pressures  by  the  monomial  formula 
are  directly  proportional  to  X3j  which  therefore  gives,  to  the 
proper  scale,  the  typical  pressure  curve.  Its  maximum  value, 
as  seen  from  the  table  of  the  X  functions,  occurs  when  x  =  0.64, 
and  its  logarithm  is  9.86390  —  10.*  Applying  this  in  equation 
(8)  gives  for  the  maximum  pressure, 

IV 

pm=  [7.69257  -  io]M—     ....     (10) 

If  the  maximum  pressure,  assumed'  to  be  the  crusher-gauge 
pressure,  is  known  by  experiment,  we  may  compute  M  from  the 
last  equation.  Thus  we  have 

,,       r  i  a  *  P™ 

M  =  [2.30743]  —  ^-        ....     (n) 

Substituting  this  in  (7)  and  (8)  we  have  while  the  powder  is 
burning, 


and 

p  =  [0.13610]  pmX,  .....     (13) 

Since  by  (12)  the  velocity  is  proportional  to  \/Xlt  this  func- 
tion represents  the  typical  velocity  curve  while  the  powder  is 

*  The  maximum  value  of  X3  occurs  when  x  =  0.6336+-  But  the  value 
of  x  given  above  is  near  enough  for  all  practical  purposes.  It  may  be 
noted  here  that  the  curve  of  X3  has  a  point  of  inflection  when  x  —  1.3891. 


102  INTERIOR   BALLISTICS 

burning.     After  the  powder  is  all  burned  the  monomial  formulas 


and 


_P^_  M'X0  ,    } 

~  ' 


are  to  be  employed. 

Example. — As  an  example  of  monomial  formulas  for  velocity 
and  pressure  take  the  following  data  from  "Notes  on  the  Con- 
struction of  Ordnance,"  No.  89,  pages  43-47: 

Gun:  8-inch  B.  L.  R.,  Model  1888.  ^  =  3617  in.3;  um 
=  205.25  in. 

Powder:  Nitrocellulose  composition,  single-perforated  grains 
of  the  following  dimensions:  length  (m)  47.69  in.;  outside  dia- 
meter (2  R)  0.4455  in.;  diameter  of  perforation  (2  r)  0.1527  in. 

From  these  dimensions  we  find  by  the  formulas  of  Chapter 

in., 

2  jf.  -  -  (0.4455  -  0-1527)  =  0-1464  in. 

*  =  ^f  =  0.0030525. 

a  =  I  +  x  =  1.0030525. 

x 
X  =  ^      ^  =  0.0030433. 

fj,  =  O. 

We  may,  therefore,  in  this  case,  assume  a  =  i  and  X  =  o 
without  material  error,  and  employ  the  monomial  formulas  (7) 
and  (8),  computing  M  either  by  (9)  or  (n),  according  as  we  take 
the  observed  muzzle  velocity  or  crusher-gauge  pressure  for  this 
purpose.  If  the  crusher-gauge  pressure  (assumed  to  be  pm)  is 
employed  equations  (12)  and  (13)  may  be  used.  If  it  is  known 
that  the  powder  is  all  burned  at,  or  near,  the  muzzle  (9)  becomes 

M  =  ^  (9') 


APPLICATIONS  103 

in  which  both  symbols  in  the  second  member  refer  to  the  muzzle. 
If  the  charge  is  not  all  consumed  at  the  muzzle  and  we  know 
the  value  of  vc  for  the  powder  used,  X0  can  be  found  by  (69), 
Chapter  IV.,  and  then  ^V  can  be  computed  by  (9).  Finally  if 
vc  is  not  known  equations  (12)  and  (13)  must  be  employed. 

As  an  example  one  shot  was  fired  with  a  charge  of  78  Ibs., 
and  a  projectile  weighing  318  Ibs.  The  observed  muzzle  velo- 
city was  2040  f.  s.,  and  crusher-gauge  pressure  30450  Ibs.  per  in.2, 
and  it  was  known  that  the  combustion  of  the  charge  was  practi- 
cally complete  at  the  muzzle.  From  the  given  data  we  find 
(taking  5  =  1.567),  A  =  0.5969,  log  a  =  0.01584,  and  z0  = 
44.548  in.  Therefore, 

um        205.25 


and  from  Table  i,  for  this  value  of  xm, 
\QgX0  =  0.77147 
logXi  =  0.41207. 
\QgX2  =  9.64060  —  10. 

.'.  log  If   =   2  log  2O40  —  0.41207    =   6.20719. 

Also  by  (61),  Chapter  IV,  log  M'  =  4.63036. 
The  equations  for  the  velocity  and  pressure  curves  for  this 
shot  are,  therefore, 

v  =  [3-10359]  V^i  .....     (16) 
and 

/>  =  [4.63036]  X8          .      .      .      .     (17). 

The  first  of  these  equations  gives,  of  course,  the  observed 
muzzle  velocity;  and  the  second  gives  (by  taking  x  =  0.64)  a 
maximum  pressure  of  31208  Ibs.  per  in.2,  exceeding  the  crusher- 
gauge  pressure  by  758  Ibs. 

If  we  determine  the  value  of  M  by  means  of  the  crusher-gauge 
pressure  we  shall  have  by  (n),  log  M  =  6.19652;  and  the 
equations  for  velocity  and  pressure  now  become 

v  =  [3.09826]  VXl 


104 


INTERIOR   BALLISTICS 


and 

p  =  [4.61969]  X3 

This  last  equation  gives  the  observed  crusher-gauge  pressure 
while  the  first  makes  the  muzzle  velocity  25  f.  s.  less  than  the 
observed.  As  muzzle  velocities  can  be  more  accurately  measured 
than  maximum  pressures,  the  first  set  of  formulas  are  probably 
the  more  accurate  and  will  be  used  in  what  follows  in  preference 
to  the  other  set. 

The  expression  for  fraction  of  charge  burned  at  any  travel 
of  projectile  is  found  from  (70),  Chapter  IV.,  and  is  for  this 
example, 

v2 

k  =  [3.02134  --  10]  Y"  .....    (l8) 

The  travel  of  projectile  is  given  by  the  equation 

u  =  z0  x  =  44.548  x  inches      ....      (19) 

The  following  table  computed  by  means  of  equations  (16), 
(17),  (18),  and  (19),  is  represented  by  the  curves  v  and  p  in 
Fig. 


i. 


X 

Travel 
U 
inches 

Velocity 
V 
ft.  sees. 

Pressure 

P 
Ibs.  per  in.2 

Fraction 
of  charge 
burned. 
k 

Pressure 
P 
Ibs.  per  in.2 

Velocity 
ft.-  sees. 

0.0 

O.O 

0.0 

00 

0.0 

84084 

0.0 

O.2 

S.QIO 

379-7 

26II5 

0.257 

65939 

749-3 

0.4 

17.819 

600.2 

30260 

0-357 

53687 

1005  .  i 

0.6 

26.729 

770-3 

3H94 

0.430 

44931 

1175.0 

0.8 

35.638 

910.1 

30949 

0.489 

38402 

1301.6 

I.O 

44-548 

1029.2 

30214 

0-539 

33369 

1401.5 

1.2 

53-457 

II33-I 

29281 

0-583 

29387 

1483-4 

i-4 

62.367 

1225.2 

28285 

0.623 

26167 

1552.3 

1.6 

71.277 

1308.0 

27288 

0.659 

23519 

1611.5 

1.8 

80.186 

1383-2 

26322 

0.692 

21306 

1663.1 

2.0 

89.096 

1452.0 

25401 

0.722 

19434 

1708.6 

2-5 

111.370 

1602.2 

23322 

0.790 

15823 

1802.8 

3-0 

I33-644 

1729.3 

21548 

0.849 

13242 

1877.0 

3-5 

I55-9I8 

1839.1 

20034 

0.901 

II3I8 

1937-5 

4.0 

178.192 

1936.0 

18733 

0.948 

9834 

1988.2 

4-5 

200  .  466 

2022  .  5 

17606 

0.991 

8661 

2031.5 

4.6073 

205.250 

2040  .  o 

17384 

1.  000 

8441 

2040  .  o 

APPLICATIONS 


I05 


The  last  two  columns  in  the  table  represented  by  the  curves 
V  and  P,  Fig.  i,  show  the  velocity  and  pressure  upon  the  supposi- 
tion that  the  powder  was  all  converted  into  gas  at  the  tempera- 


FlG    I. 

ture  of  combustion  before  the  projectile  had  moved.     They 
were  computed  by  the  formulas 

V*  =  V*X2  =  [6.97866]*,     ....     (20) 
and 

[4.92471] 


_ 
' 


,  . 


The  force  of  the  powder  (/)  and  the  velocity  of  combustion 
in  free  air  (vc)  for  this  particular  charge  and  brand  of  powder 
can  now  be  computed  by  equations  (64)  and  (67),  Chapter  IV. 
We  find/  =  1396.9  Ibs.  per  in.2  and  vc  =  0.13614  in.  per  sec. 

If  we  wish  to  compute  velocities  and  pressures  in  this  8-inch 
gun  when  the  charge  varies  K  must  be  computed  by  (90)  and  M 
by  (91),  Chapter  IV.  Since  the  weight  of  projectile  is  constant 

2 

n'  is  zero;   and   for   an  8-inch  gun  we  will  assume  that  n  =  —  , 

O 


106  INTERIOR   BALLISTICS 

—this   assumption  to  be  tested  by  experiment.     With  these 
values  of  n  and  n'  we  have 


and  therefore, 

/  =  76.519  &  .....      (22) 

Substituting  this  value  of  K,  and  the  gun  and  powder  con- 
stants in  (91),  we  have 

M  =  [2.09974]  a"  wv  .....     (23) 

Also,  from  (61),  Chapter  IV, 

M'=  [2.43084]  ^      ....  (24) 

Therefore  from  (7),  (8),  and  (10), 

v  =  [1.04987]  a1  u^VX'j.   ....     (25) 


"«  X 

p  =  [2.43084]     r    ......    (26) 


and 

£w=  [2.29474]!       .....     (27) 


which  are  the  formulas  for  velocity  and  pressure  for  this  gun  and 
brand  of  powder  in  terms  of  the  weight  of  charge. 

As  an  example,  what  would  be  the  maximum  pressure  with 
a  charge  of  79^  IDS«?  We  first  find  A  =  0.6084  and  then  log  a  = 
0.00238.  We  then  have  by  (27) 

log  pm  =  2.29474  +  •£  log  79.5  -  -  log  a  =  4-5Io65 

•*•  Pm  =  324o8  Ibs.  per  in.2 

This  agrees  very  closely  with  observation. 

We  have  the  means  of  testing  the  accuracy  of  these  equations 
to  a  limited  extent,  since  there  were  four  shots  fired  with  charges 
of  70,  78,  85,  and  88  Ibs.  The  following  table  gives -the  results 
of  all  the  necessary  preliminary  calculations  for  the  four  shots 


APPLICATIONS 


107 


fired  and  also  for  two  others  "  estimated  from  prolonged  empirical 
curves."  The  data  from  the  shot  fired  with  a  charge  of  78  Ibs. 
have  been  taken  as  the  basis  of  the  calculations.  The  gun 
constants  will  be  found  on  page  102. 


AT 

MUZZLE 

CO 

A 

log  a 

log  Z0 

Ibs. 

X 

logXo 

logXt 

logX2 

60 

0.4592 

0.18742 

.70647 

4-0347 

0.74978 

0.36944 

9.61965-10 

70 

0-5357 

0.08940 

.67540 

4-3339 

0.76150 

0.39260 

9.63110 

78 

0.5969 

0.01584 

.64883 

4.6073 

0.77147 

0.41207 

9.64060 

85 

0.6505 

9.95382 

.62414 

4.8769 

0.78066 

0.42987 

9.64919 

88 

0.6734 

9-92777 

•6I3I5 

5.0018 

0.78475 

0.43770 

9.65295 

95 

0.7370 

9.86767 

.58629 

5-3210 

0.79467 

0.45663 

9.66196 

The  computed  muzzle  velocities  and  maximum  pressures  in 
the  following  table  were  obtained  (witn  the  exception  of  the 
first  two  muzzle  velocities)  by  equations  (25)  and  (27).  The 
values  of  /  were  computed  by  (22)  and  X0  by  (69),  Chapter 
IV. 


CO 

Ibs. 

log  X0 

f 

Ibs.  per 
inch2 

MUZZLE  VELOCITY, 

F.  S. 

MAXIMUM  PRESSURES, 
LBS.  PER  IN.2 

Observed 

Com- 
puted 

O.-C. 

Observed 

Com- 
puted 

O.-C. 

60 
70 

78 
85 
88 

95 

0.74265 
0.75819 
0.77147 
0.78382 
0.78931 
0.80274 

H73 
1306 

1397 
1479 
I5H 
1593 

1600 

1839 
2040 
2200 
2275 
2454 

1600 
1844 
2040 
2205 
2276 
2441 

0 

~~  5 

0 

-  5 
—   i 

+  13 

18000 
24889 
30450 
35600 
39301 
47280 

18859 
25272 
31206 
37051 
39756 
46583 

-859 
-383 
-756 
-1451 
-455 
+697 

For  the  first  two  shots  the  powder  was  all  burned  before  the 
projectile  had  reached  the  muzzle,  as  is  shown  by  the  values  of 
log  X0.  For  these  the  muzzle  velocities  were  computed  by 
equation  (14). 

It  will  be  observed  that  the  equations  by  means  of  which 
the  muzzle  velocities  and  maximum  pressures  given  in  this 
table  were  computed  depend  for  their  constants  upon  one 


io8 


INTERIOR   BALLISTICS 


measured  velocity  only,  due  to  a  charge  of  78  Ibs.  The  measured 
crusher-gauge  pressure  for  this  charge  has  not  been  made  use 
of  at  all.  The  constant  M  upon  which  all  the  other  constants 
depend  might  have  been  determined  by  equation  (n)  in  which 
the  muzzle  velocity  does  not  enter.  But  muzzle  velocities  can 
be  more  accurately  measured  than  maximum  pressures  and 
are,  therefore,  better  adapted  to  the  determination  of  ballistic 
constants.  The  computed  maximum  pressures  in  the  table  are 
probably  nearer  the  actual  pressures  on  the  base  of  the  projectile 
than  those  given  by  the  crusher  gauge.  The  agreement  between 
the  computed  and  measured  muzzle  velocities  is  all  that  could 
be  expected  from  any  ballistic  formulas. 

To  determine  the  travel  of  projectile  when  all  the  charge  was 
burned  we  take  x  by  interpolation  from  the  table  of  the  X 
functions  corresponding  to  the  values  of  log  X0.  We  then  have : 

u  =  .x  z0. 

For  the  travel  of  projectile  when  the  pressure  is  a  maximum. 
we  have,  calling  this  travel  uf, 

u'  =  0.64  z0. 

The  following  table  gives  the  values  of  u'  and  u  for  all  the 
shots: 


ft 

u' 

u 

um  —  u 

k 

Remarks 

Ibs. 

inches 

inches 

inches 

60 

32.56 

196.56 

8.69 

70 

30-3I 

201.13 

4.12 

78 

28.51 

205.25 

0.00 

1.  0000 

85 

26.94 

209.30 

-  4.05 

0.9926 

88 

26.26 

2II.I6 

-  5.91 

0.9895 

95 

24.69 

215.88 

-10.63 

0.9813 

It  will  be  observed  that  as  the  charge  increases  the  sooner 
it  exerts  its  maximum  pressure.  The  last  column  gives  the 
fraction  of  the  charge  burned  at  the  muzzle  and  shows  that 


APPLICATIONS 

approximately  the  entire  charge  for  the  series  was  consumed 
at  the  muzzle,     k  was  computed  by  the  formula 

v2 

k         =         [6.I7483-IO]        ,1          y 

«s  A2 

In  order  to  determine  the  velocity  and  pressure  curves  for 
any  given  charge  we  should  compute  M  and  M '  by  equations 
(23)  and  (24),  and  then  employ  (7)  and  (8)  as  has  already  been 
done  for  a  charge  of  78  Ibs.  For  example,  determine  the  velocity 
and  pressure  curves  for  a  charge  of  95  Ibs.  We  have,  from  (23) 
and  (24), 

log  M  =  2.09974  +  —  log  a  +  —  log  co  =  6.31864 

log  M'  =  2.43084  +  ^  log  £  -  -  log  a  =  4.80435 
Therefore 

v  =  b^spa2!  VXi 

and 

p  =  [4-80435]  X* 
are  the  equations  required. 

Example. — Suppose  the  thickness  of  web  of  the  grain  we  have 
been  considering  to  be  increased  10  per  cent.,  all  other  conditions 
remaining  the  same.  Deduce  the  velocity  and  pressure  curves 
for  a  charge  of  78  Ibs.  In  this  case  it  is  evident  that  all  the 
charge  would  not  be  burned  in  the  gun  and  that  in  consequence 
both  the  maximum  pressure  and  muzzle  velocity  would  be 
diminished. 

It  will  be  seen  from  (69),  Chapter  IV,  that,  other  things  being 
equal,  the  value  of  X0  varies  directly  with  the  web  thickness. 
Therefore  if  this  is  increased  by  10  per  cent.,  or,  what  is  the  same 
thing,  is  multiplied  by  i .  i ,  X0  will  also  be  multiplied  by  i .  i ; 
and  from  (60)  and  (61),  Chapter  IV,  M  and  Mf  will  be  divided 
by  i.i.  Therefore  (16)  and  (17)  will  in  this  case  become, 

v  =  [3.08290] 


HO  INTERIOR   BALLISTICS 

and 

P    =    [4.58897]  *8. 

These  equations  give  vm=  1945  f.  s.,  and  pm=  28371  Ibs. 
This  is  a  loss  of  95  f.  s.  in  muzzle  velocity  and  a  diminution  of 
2079  Ibs.  in  maximum  pressure.  To  determine  the  fraction  of 
the  charge  burned  at  the  muzzle,  we  have  from  (45),  Chapter  IV, 


k- 
~ 


which  gives,  by  employing  the  muzzle  velocity  just  computed, 

k  =  0.909. 

Therefore  on  account  of  the  increased  thickness  of  web,  seven 
pounds  of  the  charge  remained  unburned  when  the  projectile 
left  the  gun. 

We  may  next  inquire  what  effect  a  decrease  of  10  per  cent,  in 
web  thickness  would  have  upon  the  muzzle  velocity  and  maximum 
pressure.  In  this  case  we  must  multiply  the  original  value  of  X0 
by  0.9  and  divide  M  and  M  '  by  the  same  fraction.  We  thus  get 

log  X0  =  0.72571 
logM  =  6.25295 
log  M'  =  4.67612 

Therefore,  from  (9),  the  muzzle  velocity  in  this  case  is  found 
to  be  2040  f.  s.;  and,  by  (10),  the  maximum  pressure,  34675  Ibs. 
That  is,  the  muzzle  velocity  remains  the  same  while  the  maximum 
pressure  is  increased  by  4225  Ibs.  per  in.2  These  examples  show 
that  for  the  greatest  efficiency  (muzzle  velocity  and  maximum 
pressure  both  considered),  the  web  thickness  for  this  form  of 
grain  should  be  such  that  the  charge  is  all  consumed  at  the 
muzzle.  From  the  value  of  X0  given  above  we  find,  by  interpola- 
tion, that  x  =  3.4890;  and,  therefore,  u  =  155.43  inches.  For 
this  travel  the  above  values  of  M  and  M'  give  v  =  1936  f.  s., 
and  p  =  22294  Ibs.  The  muzzle  pressure  comes  out  8433  Ibs. 


APPLICATIONS  III 

Suppose  for  a  hypothetical  7 -inch  gun  we  assume  the  follow- 
ing data: 

d  =  o".7 
Vc  =  4,000  c.  i. 

um  =  40  calibers  =  280  inches. 
A  =  0.6. 
5  =  1.5776 
/  =  1396.9  Ibs. 
vc  =  0.13614  in.  per  sec. 
w  =  205  Ibs. 

What  muzzle  velocity  and  maximum  pressure  would  be 
obtained,  supposing  the  charge  to  be  all  consumed  at  the  muzzle; 
and  what  must  be  the  thickness  of  web? 

The  weight  of  charge  due  to  the  given  chamber  capacity  and 
density  of  loading  is  found  to  be  86.7  Ibs.  We  next  compute 
the  following  numbers  by  formulas  given  in  Chapter  IV: 

log  a  =  0.0122 1 
Iogz0    =  1.80713 

*m    =  4-3655 

log  X»=  log  Xom=  0.76269 

log  Xi=  logXlm=  0.39492 

log  JV-  7-21529  (By  (58),  Chapter  IV), 

log  M  =  6.45260     (By  (9)) 

Then  by  (7)  and  (10)  we  find 

Muzzle  velocity  =  2653  f.  s. 
and 

Maximum  pressure  =  32112  Ibs.  per  in.2 

The  muzzle  pressure,  by  (8),  is  18413  Ibs. 

The  necessary  thickness  of  web  in  order  that  the  charge  may 
all  be  consumed  at  the  muzzle,  is  0.158  inches.  The  other 
dimensions  of  the  grains  are  immaterial. 

If  the  volume  of  the  chamber  is  taken  at  3,000  c.  i.,  all  the 


112  INTERIOR   BALLISTICS 

other  data  remaining  the  same,  we  should  have  the  following 
results : 

w  =  65.03  Ibs. 
M .  V.  =  2413  f.  s. 

pm  =  28868  Ibs.  per  in.2 
M .  P.  =  14105  Ibs.  per  in.2 
2l0  =  0.152  in. 

If  vc  =  4500  c.  i.,  we  have  the  following: 

co  =  97.54  Ibs. 
M.  V.  =  2753  f.  s. 

Pm=  33574  Ibs.  per  in.2 
M .  P.  =  24496  Ibs.  per  in.2 
2  10  =  0.160  in. 

Binomial  Formulas  for  Velocity  and  Pressure. — Binomial 
formulas  pertain  to  grains  for  which  /*  is  zero  or  so  small  that 
it  may  be  neglected,  while  X  must  be  retained  on  account  of  its 
magnitude.  To  this  class  belong  all  unperforated,  long,  slender 
grains  of  whatever  cross-section,  such  as  strips,  ribbons,  cyl- 
inders, etc.  The  binomial  expressions  for  velocity  and  pressure 
for  these  grains  are 

v2=  MX,  {i  -  NX0}    ....     (28) 
and 

p  =  M'Xi{i-  NX4]      ....     (29) 

The  second  term  within  the  brackets  has  the  negative  sign 
because  X  is  always  negative  for  these  forms  of  grain. 

Methods  for  Determining  the  Constants  M  and  N. — The 
constants  M  and  N  can  be  determined  when  the  given  experi- 
mental data  are  such  that  two  independent  equations  can  be 
formed  involving  M  and  N.  These  data  may  be  either  two 
measured  velocities  of  the  same  shot  at  different  positions  in 
the  bore;  or  a  measured  muzzle  velocity  and  crusher-gauge 
pressure, — the  latter  taken  as  the  maximum  pressure.  In 


APPLICATIONS  113 

addition  to  these  all  the  elements  of  loading,  as  well  as  the 
powder  and  gun  constants,  are  supposed  to  be  known. 

First  Case. — Let  Vi  and  v2  be  two  measured  velocities  in 
the  bore  at  the  distances  u^  and  u2  from  the  origin,  which  is  the 
base  of  the  projectile  in  its  firing  position.  From  the  gun  and 
firing  constants  compute  z0  by  (56),  Chapter  IV,  and  then  Xi  and 
x2  corresponding  to  Ui  and  u2  by  the  equation 

u 

x  =  — 

Z0 

With  these  values  of  Xi  and  x2  as  arguments,  interpolate 
from  the  table  of  the  X  functions  the  corresponding  values  of 
log  X0  and  log  Xi,  distinguishing  them  by  accents.  We  then 
have  the  two  independent  equations 

r,    2  US     V    "     CT  AT     Vff    \ 

V2    =  M  A!     (i  —  J\  X   0) 

from  which  M  and  ^V  may  easily  be  determined.     For  simplicity 
let 

^  —  \  ~  I   '  ~v7^  and   &'  =  ~v7T  • 
VV       A    ,  A    0 

We  then  have  in  a  form  well  adapted  to  logarithmic  computa- 
tion 

N  T~b  (     \ 

~  (i  -  bV)  X"0  ($0) 

and 


~  X\(i  -  N  X'0)       X'\(i  -  N  X"0 

These  equations  are  equally  adapted  to  English  or  French  units. 
Second  Case. — When  the  powder  is  not  all  burned  in  the 
gun  let  vm  be  the  observed  muzzle  velocity  and  pm  the  crusher- 
gauge  pressure.     We  then  have  the  two  independent  equations 


U4  INTERIOR   BALLISTICS 

and  ((50'),  Chapter  IV), 

pm=  [9.85640  -  10]  M'  (i  -  [0.48444]  AO  .  (32) 
Substituting  for  M'  its  value  in  terms  of  M  ((61),  Chapter  IV), 
and  making,  for  English  units, 

WlPm 

c  ==  [7-68507  --  10]  ai-)pmxi 
we  have 


N"    X0- 3.051  c    "        (i  -[o.48444]  <A,  .      .     (33) 

°  \~      x       ' 

and  then  M  from  (31).  The  X  functions  in  these  last  two 
formulas  refer  to  vm.  Any  measured  velocity  within  the  bore 
before  the  powder  is  all  burned  may  be  used  instead  of  vm.  For 
French  units  the  logarithmitic  multiplier  in  the  expression  for 
c  is  [7.56404  —  10]. 

Second  Method. — If  the  powder  is  all  burned  before  the 
projectile  reaches  the  muzzle,  we  have  from  (3) 

aF2w  N 


where  Vm  is  the  muzzle  velocity  and  X2  corresponds  to  Vm.  N 
must  be  determined  either  by  a  velocity  Vi  measured  in  the 
bore  before  the  powder  is  consumed,  or  by  the  crusher-gauge 
pressure  assumed  to  be  pm.  If  by  the  former,  we  have  from  (28) 

v?  =MXl'(i  -  NX'0}. 

Substituting  M  from  (34)  in  this  equation  and  solving  for 
^V  we  have 

4XX/z;U 


In  this  equation  X'0  and  X'2  correspond  to  the  measured 
velocity  vlm  The  travel  of  projectile  to  the  point  where  all  the 
powder  is  burned  is  found  by  the  equation  X0  =  \/N  and  a 
reference  to  the  table  of  the  X  functions.  In  using  these  last 


APPLICATIONS  115 

two  formulas  N  must  first  be  computed,  and  then  M.     Equation 
(35)  is  independent  of  the  units  employed. 

If,  as  is  usually  the  case,  there  is  no  interior  measured  velocity 
available  recourse  must  be  had  to  the  crusher-gauge  pressure  pm. 
In  this  case  we  have  by  means  of  (32)  and  (34),  and  (61),  Chap- 
ter IV,  for  English  units, 


,T         r  T(  /  r  ^4 

N  =  --  [9.21453  -  loj     i  --  (i  -    L2.79937] 


awy*m 

and  then  M  by  (34).  For  metric  units  the  logarithmetic 
multiplier  within  the  braces  becomes  [2.92040]. 

Application  to  Sir  Andrew  Noble's  Experiments.  —  These 
very  important  experiments  were  made  at  the  Elswick  works, 
Newcastle-on-Tyne,  with  a  six-inch  gun.  They  are  thus  de- 
scribed by  Sir  Andrew  *:  "The  energies  which  the  new  ex- 
plosives are  capable  of  developing,  and  the  high  pressures  at 
which  the  resulting  gases  are  discharged  from  the  muzzle  of  the 
gun,  render  length  of  bore  of  increased  importance.  With  the 
object  of  ascertaining  with  more  precision  the  advantages  to  be 
gained  by  length,  the  firm  to  which  I  belong  has  experimented 
with  a  six-inch  gun  of  100  calibers  in  length.  In  the  particular 
experiments  to  which  I  refer,  the  velocity  and  energy  generated 
has  not  only  been  measured  at  the  muzzle,  but  the  velocity  and 
pressure  producing  this  velocity  have  been  obtained  for  every 
point  of  the  bore,  consequently  the  loss  of  velocity  and  energy 
due  to  any  particular  shortening  of  the  bore  can  at  once  be 
deduced. 

"These  results  have  been  attained  by  measuring  the  velocities 
every  round  at  sixteen  points  in  the  bore  and  at  the  muzzle. 

"  Report  (1894)  on  methods  of  measuring  pressures  in  the  bore  of  guns"; 
and  "  Researches  on  Explosives,  Preliminary  Note."  An  abstract  of  these 
papers  is  given  in  the  "English  Text-book  of  Gunnery,"  1902;  in  Nature 
for  May  24,  1900;  and  in  Encyclopaedia  Britannica,  nth  edition,  article 
"Ballistics." 


Il6  INTERIOR  BALLISTICS 

These  data  enable  a  velocity  curve  to  be  laid  down,  while  from 
this  curve  the  corresponding  pressure  curve  can  be  calculated. 
The  maximum  chamber  pressure  obtained  by  these  means  is 
corroborated  by  simultaneous  observations  taken  with  crusher 
gauges,  and  the  internal  ballistics  of  various  explosives  have  thus 
been  completely  determined." 

The  velocities  at  the  sixteen  points  in  the  bore  were  deter- 
mined by  registering  the  times  at  which  the  projectile  passed 
these  points.  The  registering  apparatus  is  thus  described  by 
Sir  Andrew  in  the  "  Report,"  page  1 1 :  "  The  chronograph  which 
I  have  designed  for  this  purpose  consists  of  a  series  of  thin  disks 
made  to  rotate  at  a  very  high  and  uniform  velocity  through  a 
train  of  geared  wheels.  The  speed  with  which  the  circumference 
of  the  disks  travels  is  between  1200  and  1300  inches  per  second, 
and,  since  by  means  of  a  vernier  we  are  able  to  divide  the  inch 
into  thousandths,  the  instrument  is  capable  of  recording  the 
millionth  part  of  a  second. 

"The  precise  rate  of  the  disk's  rotation  is  ascertained  from 
one  of  the  intermediate  shafts,  which,  by  means  of  a  relay, 
registers  the  revolutions  of  a  subsidiary  chronoscope,  on  which, 
also  by  a  relay,  a  chronometer  registers  seconds.  The  subsidiary 

chronoscope  can  be  read  to  about  the th  part  of  a  second. 

5000 

"The  registration  of  the  passage  of  the  shot  across  any  of  the 
fixed  points  in  the  bore  is  effected  by  the  severance  of  the  primary 
of  an  induction  coil  causing  a  spark  from  the  secondary,  which 
writes  its  record  on  prepared  paper  gummed  to  the  periphery  of 
the  disk.  The  time  is  thus  registered  every  round  at  sixteen 
points  of  the  bore. 

"I  have  ascertained  by  experiment  that  the  mean  instru- 
mental error  of  this  chronoscope,  due  chiefly  to  the  deflection  of 
the  spark,  amounts  only  to  about  three  one-millionths  of  a 
second.  Usually  the  pressures  were  deduced  from  the  mean  of 
three  consecutive  rounds  fired  under  the  same  circumstances." 


APPLICATIONS 


The  following  table  gives  the  recorded  experimental  data 
for  the  various  kinds  of  smokeless  powders  employed  at  the 
Elswick  firings,  and  which  will  be  used  in  the  following  dis- 


cussions. 


MEASURED  VELOCITY  WHEN  PROJEC- 

Weight 

Density 

Crusher- 

TILE  HAD  TRAVELLED 

Kind  of  Powder 

of 
Charge. 
Ibs. 

Loading 

Pressure, 
Ibs. 

16.6  ft. 

21.6  ft. 

34-1  ft. 

46.6  ft. 

f.  s. 

f.  S. 

f.  S. 

f.  s. 

Cordite,  o"-4  .... 

27-5 

0-55 

47040 

2794 

2940 

3166 

3284 

Cordite,  o".35  .  .. 

22.  O 

0.44 

30352 

2444 

2583 

2798 

2915 

Cordite,  o"-3  .... 

2O.  O 

0.40 

36960 

2495 

2632 

2821 

2914 

Ballistite,  o".3  .  .  . 

20.0 

0.40 

33936 

2416 

2537 

2713 

2806 

The  cordite  used  in  these  experiments  contained  37  per  cent. 
of  gun-cotton,  58  per  cent,  of  nitro-glycerine,  and  5  per  cent,  of 
a  hydrocarbon  known  as  vaseline.  The  ballistite  was  nearly 
exactly  composed  of  50  per  cent,  of  dinitrocellulose  (collodion 
cotton)  and  50  per  cent,  of  nitro-glycerine. 

DISCUSSION  OF  THE  DATA  FOR  CORDITE,  o".4  DIAMETER.  — 

The  form  characteristics   of   cordite   are  a.  =  2,  \  =  —  —  and 
ju  =  o.     The  equations  for  velocity  and  pressure  are  therefore, 

V2=  MXi(i  -  NX0) 
and 


Since  the  second  terms  within  the  parentheses,  which  contain 
X,  have  been  made  negative,  X  in  subsequent  calculations  must 
be  regarded  as  positive. 

For  the  preliminary  calculations  we  have  the  following 
data: 

co  =  27.5  Ibs. 
w  =  100  Ibs. 

A  =  0.55 
5  =  1.56 


n8 


INTERIOR   BALLISTICS 


From  these  data  we  find  by  the  proper  formulas: 

log  a  =  0.07084 

Iogz0=  0.42178  .'.  z0=  2.6411  ft. 

To  determine  whether  the  powder  was  all  burned  in  the  gun, 
the  following  table  is  formed  which  explains  itself: 


u 

ft. 

u 

*" 

V 

(observed) 
f.  s. 

log  v  "* 

logX2 

log  V,» 

log  Fa 

vt 

16.6 

6.2853 

2/94 

6.89245 

9.68496-10 

7.20749 

3-60374 

4015 

21.6 

8.1784 

2940 

6.93669 

9.71799 

7.21870 

3-60935 

4068 

34-1 

12.9112 

3166 

7.00102 

9.76657 

7-23445 

3.61722 

4142 

46.6 

17.6446 

3284 

7.03281 

9.79440 

7.23841 

3.61920 

4161 

The  increase  of  Vi  as  shown  in  the  last  column,  indicates 
that  the  powder  was  all  burned  in  the  gun  and  between  u  = 
34.1  and  u  =  46.6  ft. 

We  will  compute  M  and  N  by  means  of  the  measured  muzzle 
velocity  (Vm)  and  the  mean  crusher-gauge  pressure  (pm),  as  these 
data  can  always  be  obtained  without  sacrificing  a  gun. 

For  cordite  equations  (36)  and  (34)  become 


N  =  [9.21453  -  10]     i  -  (i  -  [*.79973l  1j 


and 


4  V 


(36') 


(340 


The  numbers  to  be  used  in  these  formulas  are 

w  =  27.5  Ibs. 

w  =  100  Ibs. 

pm=  47040  Ibs.  per  in.2 
Fm=3284f.  s. 

logJ\T2=  9.79440  -  10  (at  muzzle) 
log  a  =  0.07084 


APPLICATIONS  IIQ 

Performing  the  operations  indicated  in  (36')  and  (34')  we 
have 

log  N  =  8.73599  -  10 
log  M  =  6.57646 

Also,  by  (62),  Chapter  IV, 

log  M'  =  4.89496. 

The  equations  for  the  velocity  and  pressure  curves  while 
the  powder  is  burning  are,  therefore, 

v-=  [6.57646]  Xj i  -  [8.73599  -  io]X0}         .     (37) 
and 

p  =  [4-89496]  *i  U  --  [8.73599  -  10]  ^4J  .      .     (38) 
After  the  powder  is  burned  we  have  from  (34'),  dropping  the 
subscript  from  Vmj  and  reducing, 

V  =  [3.6i92o]\/Ar2 (39) 

Also  from  (31)  and  (63),  Chapter  IV, 
P      [5.07979] 

•(i  +  *)* (4) 

To  determine  the  travel  of  projectile  to  the  point  where  the 
powder  was  all  burned  u  we  have,  for  cordite, 

X°~  2N 

Therefore 

logX0  =  0.96298; 

and  by  interpolation  from  the  table  of  the  X  functions, 

x  =  16.018. 
Whence 

u  =  x  z0=  42.30  ft. 

The  velocity  v  may  be  computed  by  either  of  equations  (3  7) 
and  (39),  as  they  both  give  the  same  value  to  v}  namely, 

v  =  3261  f.  s. 


120 


INTERIOR  BALLISTICS 


It  will  be  seen  that  the  increase  of  velocity  from  u  =  42.26  ft. 
to  u  =  46.6  ft.,  a  travel  of  4.34  ft.,  is  only  23  f.  s. 

Since  the  vanishing  surface  of  a  grain  of  cordite  is  zero, 
(38)  and  (40)  give  the  same  value  to  ~p.  We  find  by  either 
equation, 

p  =  2745  Ibs.  per  in.2 

The  muzzle  pressure  by  (40)  is  2431  Ibs.  The  distance 
travelled  by  the  projectile  at  point  of  maximum  pressure  is 

0.45  X  2.64  =  1.19  ft.  =  2.38  calibers. 

Equations  (37)  to  (40)  give  all  the  information  that  was 
obtained  by  Noble's  experiment  with  cordite,  o".4.  The  only 
question  that  can  arise  is  as  to  their  accuracy  in  giving  the 
velocity  and  pressure  at  every  point  of  the  bore.  Equation  (38) 
gives  the  observed  maximum  pressure  and  (37)  the  correspond- 
ing velocity.  Equation  (39)  gives  the  observed  muzzle  velocity 
and  (40)  the  corresponding  pressure.  These  equations  may  be 
further  tested  by  computing  the  velocities  for  1 6. 6,  21.6,  and 
34.1  feet  travel  and  comparing  with  the  measured  velocities. 
The  following  table  shows  the  results  of  this  procedure.  The 
differences  between  the  measured  and  computed  velocities  are 
in  all  cases  less  than  the  probable  error  in  measuring  them,  and 
are  entirely  negligible. 


Travel  of 
Projectile 

Observed 
Velocity 

Computed 
Velocity 

O.-C. 

Remarks 

1  6.6  ft. 

2794  f.  s. 

2781  f.  s. 

13 

21.6 

2940 

2942 

—  2 

34-1 

3166 

3172 

-6 

46.6 

3284 

3284 

0 

The  limiting  velocity,  Ft,  is  4161  f.  s. 

It  only  remains  to  compute  the  characteristics  vc  and  /  to 
solve  completely  the  problem  pertaining  to  this  round.     These 


APPLICATIONS  121 

are  found  to  be 

vc  =  0.38  inches  per  second, 
and 

/  =  2266  Ibs.  per  in.2 

This  value  of  /  would  mean,  if  the  problem  under  considera- 
tion were  completely  solved,  that  one  pound  of  the  gases  of  this 
powder,  at  temperature  of  combustion,  confined  in  a  volume  of 
one  cubic  foot,  would  exert  a  pressure  of  2,266  pounds  per 
square  inch.  But  the  problem  is  very  far  from  being  solved 
rigorously.  In  the  deduction  of  equation  (18),  Chapter  II, 
which  is  the  basis  of  all  our  formulas,  there  were  neglected  the 
following  energies: 

1.  The  heat  lost  by  conduction  to  the  walls  of  the  gun. 

2.  The  work  expended  on  the  charge,  on  the  gun  and  carriage, 
and  in  giving  rotation  to  the  projectile. 

3.  The  work  expended  in  overcoming  passive  resistances, 
such  as  forcing,   friction  along  the  grooves,  the  resistance  of 
the   air,    etc.      In   short   the    entire   work   of   expansion   was 
supposed  to  be  employed  in  giving  motion  of  translation  to 
the  projectile,  and  to   be  measured  by  the  acceleration  pro- 
duced. 

It  may  be  seen,  however,  from  a  careful  consideration  of 
equation  (18)  and  the  use  made  of  it  in  deducing  the  X  functions 
that  these  functions  are  independent  of  the  value  of/;  and  that 
when  this  factor  has  been  determined  so  as  to  satisfy  completely 
such  experiments  as  we  have  just  been  considering,  these  neglect- 
ed energies  are  practically  allowed  for.  Indeed,  they  are  all 
contained  implicitly  in  the  factors  M  and  N.  Similar  remarks 
apply  to  T  whose  deduced  value  from  (2),  Chapter  IV,  depends 
upon  the  exponent  of  p0/p,  about  which  there  is  considerable 
uncertainty.  But  these  characteristics  are  unnecessary  for 
determining  the  equations  of  the  velocity  and  pressure  curves 
from  such  data  as  we  have  been  considering.  But  they  are  of 


122  INTERIOR   BALLISTICS 

use  in  deducing  the  circumstances  of  motion  when  the  charge 
varies,  as  has  been  already  shown. 

We  will  now  give  a  few  illustrative  examples  which  can  be 
solved  by  this  one  round. 

Example  i. — What  thickness  of  layer  was  burned  from  the 
grains  when  the  projectile  had  travelled  16.6  ft.? 

Combining  equations  (12)  and  (14),  Chapter  IV,  gives 


X0 

That  is,  the  thickness  of  layer  burned  from  the  surface  of  a 
grain  of  powder  of  whatever  shape,  in  the  bore  of  a  gun,  varies 
directly  as  the  function  X0.  In  this  example,  applying  the  known 
values  of  10,  X0  and  X0  (for  u  =  16.6),  we  find 

/  =  0.1443  in.     Ans. 

Example  2. — What  was  the  velocity  of  combustion  of  the 
grains  at  the  point  of  maximum  pressure? 
We  have  (equation  (53'),  Chapter  IV), 

P 


=  '°'3795  B  "47  m.  per  sec.    Ans. 


Example  3.—  What  must  be  the  diameter  of  the  grains  of 
this  powder  in  order  that  the  charge  of  27.5  Ibs.  should  all  be 
burned  when  the  projectile  has  travelled  16.6  ft.? 

When  the  only  variation  in  the  charge  and  conditions  of 
loading  is  in  the  thickness  of  web,  equation  (53),  Chapter  IV, 
shows  that  the  X0  function  of  the  distance  travelled  by  the  pro- 
jectile when  the  powder  is  all  burned  is  directly  proportional  to 
the  web  thickness.  We  therefore  find,  by  employing  known 
numbers, 

2  10=  0.2885  m-     Ans. 

Cordite,  o".35.—  Preliminary  calculations  show  that  the  cor- 
dite fired  in  this  round  was  not  quite  all  burned  in  the 


APPLICATIONS  123 

We  will  therefore  compute  N  and  M  by  (30)  and  (31)  with  the 
following  data: 

log      a  =  0.21265 

log     z0  =  0.46666     .*.  z0=  2.929  ft. 

zjj  =  2583  f.  s.  =  velocity  for  21.6  ft.  travel. 
z;2  =  2915  f.  s.  =  muzzle  velocity, 
log  X'0  =  0.84614 
log  X"  0  =  0.96201 
log  X\  =  0.55164. 
log^  =  0.74763 

These  numbers  give 

log  N  =  8.73582  —  10 
logM  =  6.48164 
logMl=  4-755I4 
The  results  of  further  calculations  for  this  round  are: 

Pm=  34093 
u  =  46.96  ft. 
/  =  2277  Ibs. 
vc  =  0.315  in.  per  sec. 

The  differences  between  the  observed  and  computed  veloc- 
ities for  u  =  16.6  ft.  and  u  =  34.1  ft.,  are,  respectively,  12  and 
6  f.  s. 

The  limiting  velocity  for  this  round  is  3731  f.  s. 

The  mean  crusher-gauge  pressure  was  30352  Ibs.,  which 
is  certainly  erroneous.  The  force  of  the  powder  is  practically 
the  same  as  for  cordite,  0^.4;  but  this  latter  seems  to  be  a  quicker 
powder  than  cordite  o".35. 

Cordite,  o".3. — The  cordite  fired  with  this  charge  was  all 
burned  in  the  gun  and  we  will,  therefore,  compute  N  and  M 
by  equations  (35)  and  (34),  with  the  following  data: 

log  a  =  0.26928 

log  z0=  0.48190     . ' .     z0=  3.033  ft. 


I24  INTERIOR   BALLISTICS 

v,=  2495  f.  s. 

vm=  2914  f.  s. 
log  X2=  9.78256  —  10 
\ogX'2=  9.66596  —  10 
\ogXf0  =  0.79917 

By  means  of  these  numbers  the  various  formulas  give 
log  N  =  8.80138  -  10 
log  M  =  6.54986 
log  If  =  4.80822 
log  P' '  =  4.92766 
logFi2=  7.14642 
log  X0=  0.89759 

x_  =  10.319 

u  =  31.3  ft. 

v  =  2787.4  f.  s. 

?  =  333 1. 2  Ibs.  per  in.2 

/  =  2521  Ibs. 

vc  =  0.309  in.  per  sec. 

The  computed  maximum  pressure  is  37287  Ibs.,  which  is  but 
327  Ibs.  in  excess  of  the  mean  crusher-gauge  pressure.  The 
corresponding  velocity  is  877.2  f.  s.,  and  travel  of  projectile 
1.33  ft.  The  differences  between  the  observed  and  computed 
velocities  for  u  =  21.6  ft.,  and  u  =  34.1  ft.  are  i  and  5  f.  s. 
respectively. 

The  limiting  velocity  is  3743  f.  s. 

This  powder  is  apparently  stronger  than  either  the  0^.35  or 
o".4  cordite.  These  latter  are  evidently  of  the  same  composi- 
tion, known  as  "Mark  i,"  while  the  former  may  have  been  the 
so-called  "Cordite  M.  D.,"  which  is  said  to  have  a  slightly 
reduced  rate  of  burning  and  to  give  higher  velocities.  Its 
composition  is  gun-cotton  65  per  cent.,  nitro-glycerine  30  per 
cent.,  and  mineral  jelly  5  per  cent. 

Example.— Suppose  the  cordite,  o".3,  to  be  moulded  into 


APPLICATIONS  125 

cubes  of  the  same  web  thickness.  Determine  the  equations  of 
velocity  and  pressure.  We  have  a  =  3,  X  =  —  i,  and  ju  =  — , 
while  FI  and  X0  remain  the  same  as  already  found.  We  now 

have  M  =   — 1  ,  N  =  —  =-,  and  N'  =  —-=-;:    The  equations  are 
X0  X0  3  X0" 

therefore, 

ir  ••=  [6.72595]  Xj(i  -  [9.10241  -  10]  X0+  [7.72770  -  10]  X*  } 

and 

p  =  [4.98431]  X*{i  -  [9.10241  -  io]X4  +  [7.72770  -  10]  Y,  } 

The  maximum  pressure  computed  by  this  last  formula  is  45726 
Ibs.  The  muzzle  velocity  is,  of  course,  the  same  as  before,  as 
is  also  the  velocity  v. 

Application  to  the  Hotchkiss  57  mm.  Rapid-Firing  Gun. — 
The  data  for  the  following  discussion  are  taken  from  a  paper  by 
Mr.  Laurence  V.  Benet,  printed  in  the  Journal  U.  S.  Artillery, 
Vol.  i,  No.  3.  The  gun  experimented  with  was  a  standard 
pattern,  all  steel,  57  mm.  Hotchkiss  rapid-firing  gun,  and  the 
experiments  consisted  in  "cutting  off  successive  lengths  from  the 
chase  and  observing  the  velocities  of  a  series  of  rounds  fired  with 
each  resulting  travel  of  projectile."  The  data  necessary  for 
this  discussion  are  the  following: 

GUN  DATA. 

Area  of  cross-section  of  bore,  0.2592  dm.2 

Equivalent  diameter,  5.745  cm. 

Net  volume  of  powder  chamber,  0.887  dm.3 

POWDER    AND    PROJECTILE. 

"Two  brands  of  the  same  type  of  smokeless  powder  were 
employed,  both  of  which  were  manufactured  at  the  Poudrerie 
Nationale  de  Sevran-Livry;  they  were  designated  as  B  NI 


126 


INTERIOR  BALLISTICS 


and  B  N1M.  These  powders  are  in  the  form  of  thin  strips,  which 
are  scored  longitudinally  on  one  side  with  a  series  of  parallel  and 
very  narrow  grooves.  The  chemical  composition  is  unknown." 
The  grains  were  of  the  following  dimensions  and  densities: 


BN,. 

76  mm. 
1.4  mm. 
0.5  mm. 


B  1 

85 
1.6 
0.6 
1.78 


mm. 
mm. 
mm. 


Length  of  strips, 
Distance  between  scores, 
Thickness  of  strips, 
Specific  gravity, 

The  elements  of  loading  were  as  follows: 

Weight  of  charge,  0.460  kilos.  0.400  kilos. 

Weight  of  projectile,  2.720  kilos.  2.720  kilos. 

Density  of  loading,  0.519  0.451 

The  velocities  were  measured  by  means  of  two  Boulenge"- 
Breger  chronographs  on  independent  circuits;  and  the  pressures 
were  determined  by  means  of  a  crusher  gauge  seated  in  the  breech 
block  of  the  gun.  The  mean  pressures  at  the  breech  were  for 
B  Ni  powder,  2547  kilos,  per  cm.2;  and  for  B  N144,  2543  kilos, 
per  cm.2 

From  the  firing  records  was  obtained  the  following  table 
giving  the  velocity  of  the  projectile  corresponding  to  each  length 
of  travel  in  the  bore: 


Velocity  in 

TRAVEL  < 

DF  SHELL 

Velocity  in 

Bore  with 

Bore  with 

Remarks 

BNr 

Metres 

Calibers 

£N144 

Metres  per  Sec. 

Metres  per  Sec. 

543-1 

0.880 

15-44 

503.7 

574-4 

.051 

18.44 

534-9 

595-0 

.222 

21.44 

553-1 

612.6 

•393 

24.44 

565-0 

622.3 

•564 

27.44 

573-5 

636.5 

.792 

31-44 

591.0 

648.3 

2.020 

35-44 

600.7 

We  will  first  consider  the  powder  B  Ni,  and  compute  by  the 


APPLICATIONS 


127 


proper  formulas — already  many  times  referred  to, — the  values 
of  a,  z0  and  x  for  the  given  charge  and  travels  of  projectile. 
Then  take  from  the  table  of  the  X  functions  the  logarithms  of 
A"0,  X}  and  X2.  All  these  are  given  in  the  following  table  for 
convenient  reference: 

log  a  =  0.11053.       Iogz0=  9-35962  -  io- 


M 

Metres 

X 

logXo 

log  X, 

log  X2 

Remarks 

0.880 
I.05I 
1.222 

1-393 
1.564 
1.792 
2.O20 

3-8447 
4.5918 

5.3389 
6.0860 
6.8362 
7.8293 
8.8254 

0.74183 
0.77092 
0.79521 
0.81604 
0.83425 
0.85540 
0.87382 

0.35356 
0.4IIOO 

045765 
0.49670 

0.53013 
0.56819 
0.60062 

9.6II73 
9.64008 
9.66244 
9.68067 
9.69590 
9.71279 
9.72681 

It  is  known  that  the  powder  was  all  burned  in  the  gun,  as 
might  be  also  inferred  from  the  thinness  of  web;  and  the  first 
step  is  to  determine  the  travel  of  projectile  when  this  takes 
place,  in  other  words  the  value  of  u.  On  account  of  the  "series 
of  parallel  and  very  narrow  grooves  "  with  which  the  strips  were 
scored  on  one  side,  it  is  difficult  to  ascertain  the  form  character- 
istics from  geometrical  considerations.  Their  determination  will 
therefore  be  left  until  M  and  N  are  computed  from  the  measured 
velocities.  /*  will  be  considered  zero. 

The  expression  for  F\,  the  limiting  velocity,  is 


where  V  is  any  velocity  after  the  powder  is  all  burned  and 
X2  a  function  of  the  corresponding  travel  of  projectile.  If  then 
we  compute  Vl  by  this  formula  for  all  the  measured  velocities, 
and  find  that  it  is  approximately  constant  for  a  certain  num- 
ber of  measured  velocities  nearest  the  muzzle,  we  shall  have 
an  indication  of  the  travel  of  projectile  when  the  powder 


128 


INTERIOR   BALLISTICS 


is  all  burned.     The  following  table  gives  the  values  of 
computed: 


so 


u 

Fi 

0.880  m. 

849  m.  s. 

.051 

869 

.222 

878 

•393 

885 

•564 

883 

.792 

886 

2.020 

888 

An  examination  of  this  table  shows  that  u  lies  between  1.222 
and  1.393,  or  that  x  lies  between  the  numbers  5.3389  and  6.0860 
and  rather  nearer  the  former  than  the  latter. 

We  will  assume  x  =  5-6  and  Vi  =  885.5  m-  s->  which  is  a 
mean  of  the  last  four  values.  Since 


we  find 

v=  605.1  m.  s. 

N  and  M  can  now  be  computed  by  (30)  and  (31),  which  do 
not  contain  the  form  characteristics. 

The  data  are:  ^=  543.1  m.  s.,  v2  =  605.1  m.  s.,  logX'0  = 
0.74183,  log  *",=  0.80284,  log  A"!  =0.35356,  and  logX'\  = 
0.47205. 

The  results  of  the  calculations  are: 

log   N  =  8.68636  -  10   1 

log  M  =  5.25171  I  While  powder  is  burning. 

log  M'  '  =  3.62063 

F  =  [2.94719]  VX2} 

_  [3.7^589  I  After  powder  is  all  burned. 

(*  +  *)*  J 

The  following  table  shows  the  agreement  between  the  ob- 
served and  computed  velocities: 


APPLICATIONS 


129 


i 

Travel 

VELOCITIES 

of 

o.-r. 

Remarks 

Projectile 

Observed 

Computed 

O.88O  m. 

543.1  m.  s. 

543.1  m.  s.                o.o 

1.051 

5744 

572-8 

1.6 

1.222 

595-0 

597-4 

-2.4 

1-393 

612.6 

613.1 

-0.5 

1.564 

622.3 

623.9 

-1.6 

1.792 

636.5 

636.3 

0.2 

2.O20 

648.3 

646.5 

1.8 

The  greatest  of  these  differences  is  less  than  one-half  of  one 
per  cent,  of  the  observed  velocity  and  the  others  are  practically 
nil.  The  maximum  pressure  computed  by  the  formula 

pm=  [9.85640  -  10]  M'{i  -  [0.48444]  N} 

is  2555  kilos,  per  cm.2,  differing  by  less  than  one- third  of  one 
per  cent,  of  the  mean   crusher-gauge  pressure.     These  results 
show  that  the   assumed  value  of  x=  5.6  is  practically  correct. 
Finally,  we  have, 

/  =  7883  kg.  per  cm.2 
and 

vc=  0.438  cm.  per  sec. 
-  0.172  in.     "     " 

The  form  characteristics  a  and  X  can  be  computed  by  the 
formulas 

MX. 


V? 


and  X  =  N  X, 


From  these  we  find  a.  =  1.4460  and  X  =  0.3045. 

For  the  B  $144  powder  the  equations  for  the  velocity  and 
pressure  curves  are  found,  by  a  process  entirely  similar  to  the 
above,  to  be,  —  while  the  powder  is  burning,  — 

v2  =  [5-24187]  X,  (i  -  [8.75166  -  10]  X0) 
and 

p  =  [3-56310]  ^3  (i  -  [8.75166  -  10]  X<) 
9 


130 


INTERIOR   BALLISTICS 


After  the  powder  is  burned  the  equations  become 
V  =  [2.92007]  VX2 


[3-68425] 


and 


The    following    table    shows    the    agreement    between    the 
observed  and  computed  velocities: 


Travel 

Observed 

Computed 

of  Projectile 

Velocity 

Velocities 

o.-c. 

Remarks 

m. 

m.  s. 

m.  s. 

0.880 

5037 

503.9 

—  O.2 

I.05I 

534-9 

532.2 

2.7 

1.222 

553-1 

553-6 

-0-5 

1-393 

565-0 

566.0 

—  1.0 

1.564 

573-5 

576.5 

-3-o 

1.792 

591.0 

588-4 

2.6 

2.020 

600.7 

598.5 

2.2 

The  value  of /for  B  Nni  comes  out  8001.5  kilos,  per  cm.2,  and 
DC  is  found  to  be  0.5268  cm.  per  sec.  This  powder  is  therefore 
slightly  "stronger"  than  B  NI  and  about  22  per  cent,  quicker; 
and  this  notwithstanding  its  greater  density. 

Application  to  the  Magazine  Rifle,  Model  of  1903. — The 
following  data  pertaining  to  this  rifle  were  obtained  partly 
from  a  descriptive  pamphlet  issued  by  the  Ordnance  Depart- 
ment, and  partly  through  the  courtesy  of  officers  of  the 
Ordnance  Department  on  duty  at  the  Springfield  Armory  and 
Frankford  Arsenal,  to  whom  the  writer  is  under  special  obliga- 
tions : 

Caliber,  0.3  inches. 

Volume  of  chamber,  0.252  cubic  inches. 

Total  travel  of  bullet  in  bore,  22.073  inches. 

Mean  weight  of  powder  charge,  44  grains. 

Weight  of  bullet,  220  grains. 

"The  standard  muzzle  velocity  of  this  ammunition  is  2300 


APPLICATIONS  131 

f.  s.,  with  an  allowed  mean  variation  of  15  f.  s.  on  either  side  of 
the  standard.  The  powder  pressure  in  the  chamber  is  about 
49,000  pounds  per  square  inch." 

The  powder  used  with  this  rifle  is  composed  essentially  of 
70  per  cent,  nitrocellulose  and  30  per  cent,  nitro-glycerine.  "The 
grains  are  tubular,  being  formed  by  running  the  powder  colloid 
through  a  die  0.09  inch  in  diameter,  with  a  pin  0.03  inch  in 
diameter;  and  the  string  thus  made  is  cut  21  to  the  inch." 
There  are  considerable  variations  in  the  length  and  diameter  of 
the  grains  "due  to  the  fact  that  the  string  is  not  cut  exactly 
perpendicular  to  its  axis,  and  to  irregularities  in  shrinking. 
There  are  83,000  to  91,000  grains  per  pound.  The  specific 
gravity  is  about  1.65,  and  the  gravimetric  density  is  from  0.90 
to  0.94." 

On  account  of  the  tubular  form  of  the  grains  the  character- 
istic M  is  zero,  and  therefore  the  equations  for  velocity  and 
pressure  are  binomials.  We  have  reliable  measured  in- 
terior velocities  for  this  rifle,  obtained  at  the  Springfield 
Armory  in  the  fall  of  1903,  by  firing  with  a  rifle  the  barrel 
of  which  was  successively  cut  off  one  inch.  Five  shots  (some- 
times more)  were  fired  for  each  length  of  barrel  and  the 
velocities  were  measured  at  a  distance  of  53  ft.  from  the 
muzzle,  and  reduced  to  muzzle  velocity  by  well-known  methods. 
(See  Table  A.) 

It  is  known  that  the  charge  in  the  magazine  rifle  is  all  burned 
at,  or  very  near,  the  muzzle.  We  may,  therefore,  take  the  two 
extreme  reduced  velocities  of  the  series  for  Vi  and  v2  and  thereby 
minimize  the  effects  of  errors  in  measuring  the  velocities.  The 
firing  data  are  then, 

Vi=  1274  f.s.;         v2=  2277.6  f.  s. 
ui  =  3-°73  m- ;         ^2  =  20.073  m- 

The  weight  of  charge  in  these  firings  was  45.1  grains  and 
weight  of  bullet  220  grains.  The  preliminary  calculations  give 


132 


INTERIOR   BALLISTICS 


A  =  0.7077 
log  a  =  9.90686  —  10 
log  z0=  0.30878  .'.     z0=  2.036  in. 

Xl=  1.5093,         lo«X'0=  0.57969,     logX\=  0.00146. 

x2=  10.84125,    \ogX"0  =  0.90504,    \Q^X'\  =  0.65421. 

These  numbers  and  the  velocities  ^  and  vz,  substituted  in 
(30)  and  (31),  give 

log  N  =  8.73379  -  10 
and 

log  If  =  6.30896. 
We  also  find 

log  If' =  4.91902. 

The  formulas  for  velocity  and  pressure  are,  therefore, 

v*  =  [6.30896]  X,  {i  -  [8.73379  -  10]  X0] 
p  =  [4.91902]  X3  {  i  -  [8.73379  -  10]  X4} 
We  have 

MX, 


and  this  substituted  in  (15),  Chapter  IV,  gives 

y  =  £>  -     -X0[i  -  [8.73379  -  10]  Xo\ 

V2 

Since  in  this  case  v  and  X2  refer  to  the  muzzle,  we  have  for 
the  powder  burned,  in  grains, 

y  =  [0.99736]  X0  { i  -  [8.73379  -  10]  X0} 
or,  in  another  form  more  convenient  for  computation, 

v2 
y  =  [4.68840  -  10]  TT 

A2 

Table  A  gives  the  measured  and  computed  velocities  for  the 
travels  of  projectile  in  the  first  column,  and  also  the  weight  of 
powder  burned  at  each  travel. 


APPLICATIONS 


TABLE  A 


Travel 
of  Projectile, 

inches 

Mean  Velocity 
53  Feet 
from  Muzzle, 
f.  s. 

Muzzle  Velocity 
Deduced  from 
Measured 
f.  s. 

Computed 
Velocity, 

f.  s. 

o.-c. 

Powder 
Burned, 

grains 

3-073 

1253 

1274 

1274 

0 

29.99 

4-073 

1402 

1426 

1432 

-  6 

32.61 

5-073 

1531 

1558 

1555 

3 

34-63 

6.073 

1633 

1662 

1656 

6 

36.25 

7-073 

1742 

1772 

1740 

32 

37-59 

8.073 

1771 

1802 

1812 

—  10 

38.71 

9-073 

1860 

1894 

I874 

20 

39-66 

10.073 

1909 

1943 

1929 

H 

40.48 

11.073 

1957 

1992 

1976 

16 

41.19 

12.073 

1989 

2023 

2018 

5 

41.81 

13-073 

2016 

2052 

2057 

-  5 

42.36 

I4-073 

2050 

2086 

2091 

-  5 

42.83 

15-073 

2069 

2105 

2122 

-17 

43-25 

16.073 

2104 

2140 

2151 

—  ii 

43-63 

17.073 

2129 

2165 

2177 

—  12 

43-95 

18.073 

2183 

2219 

2200 

19 

44-25 

19.073 

2163 

2200 

2222 

—  22 

44-5° 

20.073 

2201 

2238 

2242 

-  4 

44-73 

21.073 

2203 

2240 

226l 

—   i 

44-93 

22.073 

2240 

'  2278 

2278 

o 

45-io 

Table  B,  on  page  134,  supplements  Table  A  by  giving  com- 
puted velocities  and  pressures  from  the  origin  of  motion.  The 
velocity  curve  in  the  diagram,  Fig.  2,  on  page  135,  shows  at  a 
glance  the  agreement  between  theory  and  observation. 

It  will  be  observed  that  the  computed  pressures  depend 
entirely  upon  two  measured  velocities.  Also  that  the  maximum 
pressure  occurs  when  x  =  0.45,  and  agrees  with  the  official 
statement.  The  muzzle  pressure  is  about  6,000  Ibs.  per  in.2 

Powder  Characteristics. — The  form  characteristics  of  these 
grains  according  to  the  given  dimensions  are 

a=  1.63  and  X  =  0.3865. 

But  these  minute  grains,  of  which  there  are  560  in  the  service 
charge,  shrink  irregularly  and  many  of  them  doubtless  are  more 
or  less  abraded  and  perhaps  broken,  so  that  it  is  impossible  to 
determine  the  mean  values  of  a  and  X  geometrically  with  any 


134 


INTERIOR  BALLISTICS 

TABLE  B 


X 

u, 
inches 

Computed 
Velocity, 

f.  s. 

Computed 
Pressure, 

Ibs.  per  inch2 

Powder 
Burned, 

grains 

Pressure  on 
Base  of 
Projectile, 
pounds 

O.OOO 

0.000 

0.000 

o.ooooo 

0.000 

OOO 

0.001 

0.002 

8.590 

4501 

1.081 

318 

O.OI 

0.020 

47.854 

13835 

3.375 

978 

O.I 

0.204 

254-91 

37008 

10.139 

2556 

0.2 

0.407 

408.98 

45117 

13.842 

3189 

o-3 

0.6II 

531.70 

48456 

16.475 

3425 

0.4 

0.814 

635.14 

49640 

18.554 

3509 

0.45 

0.916 

681.48 

49769 

19.454 

3518 

0.5 

I.OI8 

724.89 

49695 

20.283 

0.6 

1.222 

804.25 

49118 

21.766 

0.8 

1.629 

939-77 

47039 

24.221 

I.O 

2.036 

1052.5 

44492 

26.205 

1.2 

2-443 

1148.8 

41882 

27.867 

'.'.'.'. 

1.4 

2.850 

1232.6 

39369 

29.291 

1.6 

3.258 

1306.4 

37023 

30.533 

2.O 

4.072 

32854 

2.5 

5.090 

28^  SO 

3 

6.108 

•^^oo 
25061 

4 

8.144 

19812 

5 

I0.l8o 

16086 

6 

I2.2I6 

IT.  T.IQ 

7 

14.252 

OO     :/ 
III88 

8 

16.288 

9501 

9 

18.324 

8134 

10 

20.360 



7006 

ii 

22.396 

604.  S 

T^\J 

certainty.     They  may,  however,  be  deduced  from  the  values 
of  M  and  N.    From  these  we  find 


Finally  we  find 


and 


a  =  1.7710 
x  =  °-4353 

/  =  1622.5  Ibs.  per  in.2 
vc=  0.28  in.  per  second. 


Formulas  for  Designing  Guns  for  Cordite.— The  caliber,  of 
course,  is  given,  and  the  weight  of  the  projectile  of  desired  length 


APPLICATIONS 


and  form  of  head  can  be  computed  by  known  methods.*     The 
grain  characteristics  and  density  of  cordite  and  the  values  of  / 


ID  15 

Travel,  inches. 

FIG.   2. 


20      22.07 


and  vc  are  also  known.  The  necessary  formulas  for  this  discussion, 
given  in  the  order  in  which  they  will  be  used,  are  the  following:— 

&  =  [8.55783]  A  Vc (a) 


z0=  [1.54708]— £         (c) 


*  See  the  author's  "Handbook"  (Artillery  Circular  N),  chapter  xi. 


136  INTERIOR   BALLISTICS 


v:~= [4.44383]^  .   .   .   . 


.  .  .  to 

L  v  u       '    ) 

in  which 

&  _  [2.i289o]/ 

This  equation  is  deduced  from  (32),  eliminating  M'  and  N. 

2  V  2  I 

X0'         ~  2  X0  ' 
M'=  [7.82867  -  io]M^ (g) 

_   _.    __      p  ,.  _    — —    "i  /  7  \ 

#  =  M '  ^3  (i  -  N  X,) (i) 

pm=  [9.85640  -  10]  M'  (i  -  [0.48444]  N)    .      .     (j) 

10=  [8.56006  -  10]  — 


Example. — Take  the  hypothetical  y-inch  gun  already  con- 
sidered on  page  in,  for  which  d  =  jff  and  w  =  205  Ibs.  For 
cordite  of  o".3  diameter  we  found  /  =  2521  Ibs.  per  in.2,  and 
vc=  0.309  in.  per  sec.  Also  5  =  1.56.  The  only  assumptions 
necessary  are  the  volume  of  the  chamber  (Vc)  and  the  density 
of  loading.  And  this  last  is  not  purely  arbitrary,  since  considera- 
tions of  safety  to  the  gun  and  its  efficiency  restrict  its  value  to 
narrow  limits, — say  from  0.4  to  0.6.  This  latter  value  is  often 
exceeded,  especially  in  our  service;  but  it  is  believed  that  by 
choosing  the  proper  shape  and  size  of  grain  this  can  always  be 


APPLICATIONS 


137 


avoided.  As  /  and  vc  are  unusually  large  for  cordite,  we  will 
take  A  =  0.4;  and  for  a  first  assumption  will  give  the  chamber 
a  volume  of  3,000  c.  i.;  which  is  less  than  the  volume  of  the 
chamber  of  the  6-inch  wire- wound  gun.  Finally  we  will  take 
Pm=  37>oo°  Ibs.  per  in.2,  leaving  the  muzzle  velocity  and  travel 
in  the  bore  for  later  consideration. 

From  the  given  data  we  find,  by  means  of  the  above  formulas, 
w  =  43-352  Ibs-,  log#  =  0.26928,  log  z0  =  1.76317,  log  Fi2  = 
7.17066,  log  X0  =  0.90184,  log  M  =  6.56985,  log  Mf  =4.80398 
and  log  TV  =  8.79713  —  10. 

The  equations  for  velocity  and  pressure  are,  therefore, 

v*=  [6.56985]  X,  (i  -  [8.79713  -  10]  X0) 
and 

p  =  [4-80398]  X9  (i  -  [8.79713  -  10]  X4) 

This  last  equation,  which  is  the  same  as  equation  (/)  when 
x  =  0.45,  makes  pm=  37000,  thus  verifying  the  calculations. 

The  muzzle  velocity  will,  of  course,  depend  upon  where  we 
place  the  muzzle,  in  other  words  upon  the  value  adopted  for 
um.  If  we  regard  40  calibers  as  a  suitable  travel  in  the  bore,  we 
shall  have 

um  =  40  X  7"  =  28o/r 
whence 

xm=  ujz0=  4-8305 

For  this  value  of  x,  Table  I  gives 

log  X0=  0.77912 
log  Xi  =  0.42689 

and  these  in  the  above  velocity  equation  give 

vm=  2487  f.  s. 
We  find,  from  Table  I,  taking  log  X0  as  the  argument, 

x  =  10.613; 


138  INTERIOR   BALLISTICS 

and  by  (m) 

u  =  615.19  in. 
Also  by  (k) 

2/0=o".47 
and  by  (/) 

km  =  0-9394- 

That  is  94  per  cent,  of  the  charge  was  burned  at  the  assumed 
muzzle. 

If  the  maximum  pressure  is  increased  to  38,000  Ibs.,  the 
density  of  loading  and  volume  of  chamber  remaining  as  before, 
the  velocity  for  a  travel  of  280  inches  will  be  increased  to  2503 
f.  s.,  and  the  thickness  of  web,  or  diameter  of  the  grain,  will  be 
reduced  to  0^.45.  This  slight  diminution  in  the  diameter  of 
the  grain  increases  the  initial  surface  of  combustion  of  the 
charge  about  3^  per  cent.,  which  fully  accounts  for  the  increased 
maximum  pressure. 

If  we  take  A  =0.5,  Vc  =  3,000  c.  i.  and  pm  =  37,000  Ibs. 
per  in.2,  there  results  co  =  54.19  Ibs.,  v2So  =  2570  f.  s.,  and 
2/0=o".66. 

Trinomial  Formulas  for  Velocity  and  Pressure.  —  Trinomial 
formulas  occur  when  the  grains  of  which  the  charge  is  composed 
are  of  such  form  and  dimensions  that  the  form  characteristic  /* 
cannot  be  regarded  as  zero.  Spherical,  cubical,  and  multi- 
perforated  cylindrical  grains  are  of  this  kind.  For  the  first  two 
forms  mentioned  the  second  term  is  negative  and  the  third 
positive;  while  for  m.p.  grains  (those  used  in  our  service),  the 
second  term  is  positive  and  the  third  negative. 

For  spherical  and  cubical  grains  we  may  have,  before  the 
powder  is  all  burned,  the  two  independent  equations, 

-  N  X'  +- 


v,2  =  MX,"  (i  -  N  X"0  +  -N*  X'\] 


APPLICATIONS  139 

Put  for  convenience, 


X 


/f 

o 


3  (i  —  a  b)  2  c  (i  —  a) 

C  =  2(1  -ab*}X",  =  (i  -abYx7^  * 

Then  the  quadratic  equations  give,  using  the  sign  applicable 
to  this  problem, 

--^)5)  .  .  (4i) 

The  value  of  M  may  now  be  computed  by  either  of  the  above 
expressions  for  v2.  Or,  if  Vm  is  the  muzzle  velocity,  that  is, 
if  the  powder  is  all  burned  in  the  gun,  M  may  be  computed  by 
the  formula,  derived  from  (3), 

M  =  — ~— (42) 

^2 

If  the  powder  is  not  all  burned  in  the  gun  and  our  data  are 
a  muzzle  velocity  and  the  crusher- gauge  pressure  (assumed  to 
be  the  maximum  pressure),  N  may  be  computed  by  the  follow- 
ing process:  Compute  the  auxiliary  quantities  b,  c,  and  d  by 
the  formulas: 


5  =  [7.68507  -  10]  ^r^x','-  ,        bXi 

2  X.0  li  —  -^r 


d  =  -  3(I- 


Then 

(         /         j\  *i 

....     (43) 


The  functions  X4  and  X5  pertain  to  the  tabular  value  of  x 
which  gives  the  maximum  pressure.     If  we  take  this  to  be  0.45 


140 


INTERIOR   BALLISTICS 


no  material  error  will  ensue.     We  therefore  have 
log  Z4  =  0.48444 
log  ^5  =  0.93587 
The  function  X0  pertains  to  the  muzzle. 

It  should  be  remembered  that  equations  (41),  (42),  and  (43) 
are  applicable  to  cubical  and  spherical  grains  only. 

Application  to  Noble's  Experiments  with  Ballistite. — The 
ballistite  consisted  of  equal  parts  of  dinitrocellulose  and  nitro- 
glycerine and  was  in  the  form  of  cubes  0.3  of  an  inch  on  a  side. 
The  gun,  powder,  and  firing  data  are  as  follows :  d  =  6  inches, 
A  =  0.4,  6  =  1.56,  w  =  20  Ibs.,  and  w  =  100  Ibs.  From  these 
we  find  log  a  =  0.26928  and  Iogz0  =  0.48190.  .'.  z0=  3.033  ft. 
The  following  table,  which  explains  itself,  is  formed  for  convenient 
reference: 


Observed 

u, 

X=U/Z0 

Velocity 

log  X0 

log*! 

logX2 

Remarks 

ft. 

f.  s. 

16.6 

5473 

2416 

0.79917 

0.46513 

9.66596-10 

21.6 

7.121 

2537 

0.84069 

0.54I8I 

9.70112 

34-1 

11.242 

2713 

0.91049 

0.66339 

9.75289 

46.6 

15-363 

2806 

0.95685 

0.73940 

9.78256 

As  the  powder  was  not  quite  all  burned  in  the  gun  the  ex- 
treme measured  velocities  are  available  for  determining  N  and 
M  by  means  of  (41).  The  data  are  Vi  =  2416  f.  s.,  v2  =  2806 
f.  s.,  logJT0=  0.79917,  log  X"0  =  0.95685,  log  X\  =  0.46513 
and  logXi"  =  0.73940.  Substituting  these  in  (41),  we  find 
log  N  =  9.03843  -  10 

log  N'  =  7-59974  -  10 
log  M  =  6.62918 

log  if  ~  4.88754 

The  equations  for  velocity  and  pressure  are,  therefore, 
=  [6.62918]*!  {1-19.03843-10]  *0 


and  then 


£=[4.88754]  *3{  i  -[9.03843-10]  ^+[7.59974-10]  x6 


APPLICATIONS  141 

The  equation  for  velocity  will,  of  course,  give  the  observed 
velocities  for  u  =  16.6  ft.,  and  u  =  46.6  ft.  It  should  also  give 
the  observed  velocities,  if  our  method  is  correct,  for  u  =  21.6  ft. 
and  u  =  34.1  ft.,  and  indeed  for  every  point  in  the  bore  from 
the  firing  seat  to  the  muzzle.  The  velocities  computed  for  these 
intermediate  values  of  u  are  2536.5  f.  s.,  and  2710  f.  s.,  respect- 
ively, which  differ  so  slightly  from  the  measured  velocities  as 
to  be  negligible.  The  maximum  pressure,  which  occurs  in  this 
case  when  x  =  0.4,  is  39,163  Ibs.  per  in.2;  and  the  corresponding 
velocity  865.8  f.  s.,  and  travel  of  projectile  1.213  ft. 

The  distance  travelled  by  a  projectile  to  the  point  where  the 
powder  is  all  burned  is  determined  by  means  of  X0  and  the 
table  of  the  X  functions.  We  have  for  cubical  grains  for  which 
X  is  unity,  X0  =  i/N.  Therefore  for  this  example, 

log  X0  =  0.96157 

Corresponding  to  this  value  of  log  X0,  we  find,  by  interpola- 
tion from  the  table, 

x  =  15.8649,  logXi—  0.74694,  and  log  X2  =  9.78536  —  10. 

To  compute  #,  we  have  u  =  15.8649  X  3.033  =  48.12  ft. 
The  charge  was  therefore  not  all  burned  in  the  gun,  though  the 
fraction  of  the  charge  remaining  unburned  was  exceedingly 
small,  practically  zero.  To  get  an  expression  for  the  fraction 
of  the  charge  burned  for  any  travel  of  the  projectile,  we  have, 
from  (45),  Chapter  IV, 

v2  =  k  VS  X2 

But  from  (3),  for  cubic  grains, 

M 

Fi2=  —  -jj.     .'.  log  Fi2  =  7.11363  (for  this  example). 

Therefore  for  cubic  grains, 
. 
" 


MX2 


I42  INTERIOR   BALLISTICS 

which  for  this  example  reduces  to 

k  =  [2.88637  -  10]  -JJT 

Applying  different  velocities  and  the  corresponding  values 
of  log  X2  in  this  formula  it  will  be  found  that  the  charge  was 
practically  consumed  long  before  the  projectile  reached  the 
muzzle.  Indeed  nine  teen- twentieths  of  the  charge  was  burned 
when  the  projectile  had  travelled  16.6  ft. 

We  may  also  determine  k  in  terms  of  the  travel  of  projectile 
by  means  of  the  equation 

k  =  i  -  (  i  -  ^)3     (Eq.  (46),  Chapter  IV). 

Or,  if  we  wish  to  know  the  distance  travelled  by  the  projectile 
when  a  given  fraction  of  the  charge  is  burned,  we  have 


As  an  example,  suppose  k  =  -.     Then 


Y 

X     = 


Applying  the  value  of  N  found  for  this  round  and  completing 
the  calculations  it  will  be  found  that  one-half  the  charge  was 
burned  when  the  projectile  had  travelled  one  foot. 

The  expressions  for  V  and  P  for  this  round  are 

v  =  [3-55681]  vx2 

and 

[4.89487] 


The  last  two  formulas  give  the  velocity  and  pressure  upon 
the  supposition  that  all  the  powder  was  converted  into  gas  at 
temperature  of  combustion  before  the  projectile  had  started. 


APPLICATIONS 


The  initial  value  of  P  is  found  by  making  x  zero.     Whence 

pf=  78,500  Ibs.  per  in.2 
Finally  we  find 

;  =  2338  Ibs. 
and 

vc  =  0.266  in.  per  sec. 

The  following  table  was  computed  by  the  formulas  de- 
duced for  this  round  for  comparison  with  the  deductions  from 
Sir  Andrew  Noble's  velocity  and  pressure  curves.  Unfortunately 
these  curves,  as  published,  are  drawn  to  so  small  a  scale  and  are 
so  mixed  up  with  other  curves  that  it  is  difficult  to  get  the 
velocities  and  pressures  from  them  with  much  precision. 

NOTE  : — The  velocities  and  pressures  in  the  second  and  third 
columns  were  computed  by  formulas  slightly  different  from  those 
deduced  above.  But  the  differences  are  so  small  as  to  be  of  no 
account  in  the  discussion. 


X 

u 
ft. 

Computed 
Velocities, 
f.  s. 

Computed 
Pressures, 
Ibs.  per  inch2 

Pounds  of 
Powder 
Burned 

V, 

f.  s. 

P, 

Ibs.  per  in.2 

0.000 

o.ooo 

0.0 

0 

0.0 

o.o 

78500 

0.001 

0.003 

12.449 

4196 

0.716 

65.786 

78397 

O.OI 

0.030 

68.889 

12672 

2.206 

207.40 

77466 

0.05 

0.152 

221.52 

25254 

4-683 

457.78 

73557 

O.I 

0.303 

359-32 

32007 

6-357 

637.37 

69132 

0.2 

0.607 

569-36 

37680 

8.464 

875.21 

61560 

0.4 

1.213 

869.33 

39439 

10.967 

1174.0 

50122 

0.6 

1.820 

IO87.2 

37567 

12.549 

1372.5 

41948 

0.8 

2.427 

1257-7 

34846 

13.686 

1520.3 

35852 

1.0 

3-033 

1396.6 

32050 

14-557 

1637.0 

3H53 

2.O 

6.066 

1842.4 

21268 

I7-043 

1995-9 

18143 

3.0 

9.100 

2093.0 

15013 

18.225 

2192.5 

12363 

4.0 

12.133 

2257.1 

11164 

18.892 

2322.4 

9181 

5-o 

15.166 

2374.2 

8645 

19.299 

2416.9 

7200 

5-473 

16.6 

2419.0 

7742 

19-435 

2453.5 

6507 

7.121 

21.6 

2538.0 

5500 

17.742 

2554-9 

4809 

11.242 

34-1 

27IO.O 

2893 

19.979 

2711.8 

2782 

I5-363 

46.6 

2806.0 

1890     19-999 

2806.0 

1890 

The  computed  velocities  in  the  third  column  of  this  table, 
corresponding  to  the  travels  of  projectile  in  the  second  column, 


144  INTERIOR    BALLISTICS 

agree  very  well  with  those  deduced  from  Sir  Andrew  Noble's 
velocity  curve,  from  the  origin  of  motion  to  the  muzzle,  a  distance 
of  46.6  ft.  As  the  velocities  are  thus  shown  to  be  correct,  the 
pressures  in  the  fourth  column  are,  from  their  manner  of  deriva- 
tion as  given  in  Chapter  IV,  necessarily  correct  also.  That  is, 
they  correspond  to  the  energy  of  translation  of  a  hundred-pound 
projectile.  In  this  respect  they  are  more  accurate  than  the 
pressures  given  by  Sir  Andrew's  pressure  curve  which  was  de- 
rived from  his  velocity  curve  by  graphic  methods  not  sufficiently 
precise  for  the  great  accelerations  encountered  in  ballistic 
problems. 

The  writer  is  indebted  to  Colonel  Lissak,  formerly  Instructor 
of  Ordnance  and  Gunnery  at  West  Point,  for  the  accompanying 
diagram  (Fig.  3)  of  the  velocity  and  pressure  curves  whose  co- 
ordinates are  given  in  this  table.  Many  interesting  facts  may  be 
gleaned  from  an  examination  of  these  curves,  and  the  formulas 
by  which  their  coordinates  were  computed. 

The  two  velocity  curves  v  and  V  are  both  zero  at  the  origin 
but  immediately  separate,  attaining  their  greatest  distance  apart 
when  the  projectile  has  moved  but  a  short  distance.  They 
then  approach  each  other  very  gradually  and  become  tangent 
at  the  point  where  the  powder  is  all  burned, — practically  at  the 
muzzle.  Both  curves  are  tangent  to  the  axis  of  ordinates  at  the 
origin  and  parallel  to  the  axis  of  abscissas  at  infinity.  The 
pressure  curve  p  begins  at  the  origin,  attains  its  maximum  when 
the  projectile  has  traveled  about  15  inches,  changes  direction 
of  curvature  when  u  is  about  six  feet  and  meets  the  axis  of 
abscissas  at  infinity.  The  pressure  curve  P  is  convex  toward 
the  axis  of  abscissas  throughout  its  whole  extent.  It  lies  above 
the  curve  p  from  u  =  o  to  u •=  30  inches  (about),  then  passes 
below  p  and  the  two  curves  become  tangent  at  the  point  where 
the  powder  is  all  consumed.  Finally  the  areas  under  the  curves 
p  and  P  are  equal. 

Example    i. — Suppose   the   charge   in   the   example   under 


APPLICATIONS  145 

consideration  to  be  increased  from  20  to  25  Ibs.     Deduce  the 
equations  for  velocity  and  pressure. 

In  solving  this  example,  we  will  compute  the  new  constants 
M,  Mf,  N  and  Nf  by  equations  (80)  to  (83),  Chapter  IV;  and  as 
the  charge  is  increased  by  25  per  cent.,  a  new  value  of /must  be 

found  by  (90)  and  (go').     For  a  six-inch  gun  we  will  take  n  =  —  t 

o 

provisionally;  and  since  the  weight  of  the  projectile  remains  the 
same,  n'  must  be  zero.     We  therefore  have 

K  =  86.12, 

and  the  new  value  of  /  is  2518  Ibs. 

The  new  values  of  a  and  z0  for  co  =  25  Ibs.,  are 

log  a  =  0.13321 

Iogz0  =  0.44277         .*.     z0=  2.772  ft. 

Applying  these  numbers  in  the  equations  above  mentioned 
we  find  for  a  charge  of  25  Ibs., 

logM=  6.73881 
log  If  =  5.03633 
log  N  =  9.01885 
logN'  =  7.56058 

which  give  the  equations  required. 

These  constants  give  pm=  55676  Ibs.,  and  a  velocity  of 
2841  f.  s.,  for  a  travel  of  16.6  ft.  That  is,  an  increase  of  5  Ibs. 
in  the  charge  increases  the  maximum  pressure  16,800  Ibs.  per 
in.2,  and  the  velocity  at  16.6  ft.  travel,  425  f.  s.  Taking  the 
reciprocal  of  N  gives 

log  X0  =  0.98115 
and  from  the  table, 

x  =  18.1425 
and 

u  =  18.1425  X  2.772  =  50.29^. 


10 


146  INTERIOR  BALLISTICS 

The  limiting  velocity  and  fraction  of  charge  burned  are  given 
by  the  equations 


Therefore 

log  Fi2  =  7.24284 
and 

y  =  k  &  =  [4-I5510  ~  IQ]  ~£~ 

•rt-a 

From  this  last  formula  we  find  when  u  =  16.6  ft.,  y  —  24.18 
Ibs. 

The  pressure  at  this  point  is  found  to  be  10480  Ibs.  per  in.2 
It  is  interesting  to  compare  these  results  with  those  found  with 
a  charge  of  20  Ibs. 

In  order  to  lessen  the  maximum  pressure  the  grains  must 
be  increased  in  size  and  thus  diminish  the  initial  burning  surface. 
Suppose  we  increase  the  size  of  the  cubes  from  0^.3  to  0^.5  on  a 
side.  Determine  the  equations  of  velocity  and  pressure  for  a 
charge  of  25  Ibs.  An  examination  of  equations  (80)  to  (82), 
Chapter  IV,  will  show  that  when  the  only  change  in  the  data  is 
in  the  thickness  of  web  the  new  values  of  M,  M'  ,  and  N  will  be 
found  by  multiplying  the  previously  determined  values  of  these 
constants  by  the  ratio  of  the  web  thicknesses,  —  in  this  case  by 
0.6.  We  therefore  have  for  25  Ibs.  of  0^.5  cubes 

log  M  =  6.51696 
log  M'=  4.81448 
log  N  =  8.79700  —  10 
log  N'=  7.11688  -  10 

From  these  we  get 

pm  =  38437  Ibs.  per  in.2 
and 

v  =  2571  f.  s.,  for  u  =  16.6  ft. 
The  measured  velocity  for  this  travel  of  projectile  was  2416 


APPLICATIONS  147 

f.  s.,  with  a  charge  of  20  Ibs.  of  o" '.3  cubes.  Therefore  by 
increasing  the  weight  of  charge  5  Ibs.,  and  at  the  same  time 
enlarging  the  grain  from  o" '.3  to  0^.5  on  a  side  the  velocity  is 
increased  155  f.  s., — and  this  without  increasing  the  maximum 
pressure,  though  the  mean  pressure  is,  of  course,  considerably 
increased. 

The  pressure  for  u  =  16.6  ft.,  with  a  charge  of  20  Ibs.  of  the 
smaller  grains,  was  7741  Ibs.;  and  with  a  charge  of  25  Ibs.  of  the 
larger  grains,  the  pressure  for  the  same  travel  would  be  11181 
Ibs.  The  powder  actually  burned  during  this  travel  of  projectile 
is  a  little  more  in  this  latter  case  than  in  the  former,  and  the 
space  in  which  it  has  been  confined  during  its  expansion  is  less, 
both  of  which  facts  account  for  the  greater  work  performed. 

From  equation  (19'),  Chapter  III,  it  follows  that  for  two  equal 
charges  made  up  of  grains  of  the  same  form  and  differing  only 
in  their  size,  the  entire  initial  surfaces  of  the  two  charges  vary 
inversely  as  the  thickness  of  web.  Therefore  the  initial  surface 

of  the  charge  of   o.r/5  grains  is  —  of  the  initial  surface  of  the 

0 

same  charge  of  o."3  grains.  This  accounts  for  the  two  charges 
giving  the  same  maximum  pressure.  It  may  be  remarked  that 
the  same  results  would  have  been  obtained  if  the  grains  had 
been  spherical  instead  of  cubical. 

Application  to  Multiperforated  Grains. — A  peculiar  diffi- 
culty arises  in  the  application  of  any  system  of  interior 
ballistic  formulas  to  multiperforated  grains  from  the  fact  that 
they  do  not  retain  their  original  form  until  completely  con- 
sumed— as  do  all  other  forms  of  grain  in  use, — but  each  grain 
breaks  up,  when  the  web  thickness  proper  is  burned  through, 
into  twelve  slender  rods,  or  " slivers,"  which  burn  according  to 
a  different  law;  and  thus  two  independent  sets  of  formulas 
become  necessary  to  represent  what  actually  takes  place  in  the 
gun.  It  was  previously  sought  to  overcome  this  difficulty  by 
supposing  the  web  thickness  to  be  slightly  increased  so  as  to 


148  INTERIOR  BALLISTICS 

satisfy  the  equation  of  condition 

a  (i  +  X  -  M)  =  i 

and  thus  ignoring  the  slivers.*  This  method  represents  quite 
satisfactorily  the  actual  circumstances  of  motion  so  long  as  the 
grains  retain  their  original  form,  but  not  afterward.  It  assumes 
that  the  slivers  are  all  burned  with  the  fictitious  web  thickness ; 
that  is,  when,  in  all  our  guns,  the  projectile  has  performed 
approximately  half  its  travel  in  the  bore ;  while  it  is  certain  that 
in  most  cases  with  our  service  powders  they  are  not  completely 
consumed  when  the  projectile  leaves  the  bore.  It  is  necessary, 
therefore,  to  divide  the  entire  combustion  of  the  grain  into  two 
periods  and  to  deduce  formulas  that  shall  represent  the  law  of 
burning,  as  well  as  the  circumstances  of  motion,  for  each  period. 
From  equation  (22),  Chapter  III,  we  have,  for  m.p.  grains 

y  I  (  I 

k  =   ~-  =  a-r-}  I  +\~j /i- 

co  1Q  ^  i0  I 

which  gives  the  fraction  of  the  charge  consumed  when  any 
thickness  /  of  the  web  has  been  burned,  and  this  without  any 
reference  to  the  law  of  burning.  When  /  =  /Q,  that  is,  when 
the  entire  web  thickness  has  been  burned,  this  equation  becomes 

k'  =  a  (i  +  X  -  /*) 

in  which  k'  is  the  fraction  of  the  charge  less  the  slivers.  If  we 
substitute  for  a,  X  and  n  their  values  for  any  of  our  m.p.  grains, 
we  shall  find  for  this  critical  point, 

k'=  0.85  (about), 

and  therefore  the  slivers  constitute  approximately  15  per  cent, 
of  the  charge.  These  slivers  burn  according  to  another  law. 
We  may  regard  them  as  slender  cylinders  whose  form  character- 
istics are  very  approximately 

a  =  2,  X  =  —  i,  n  =  c. 

*  See  Journal  U.  S.  Artillery,  vol.  24,  p.  196,  and  vol.  26,  pp.  141  and  276. 


APPLICATIONS  149 

We  will  now  deduce  formulas  for  each  period  of  burning. 

Designate  all  symbols  referring  to  the  point  where  the  grains 
are  converted  into  slivers  by  an  accent,  and  those  relating  to 
the  muzzle — including  M,  M'  and  N — by  a  subscript  m. 

Equation  (n),  Chapter  IV,  becomes,  by  suitable  reductions, 


,  vc  \/  a  w  d> 
A'  =  :  [8.56006  —  loj -7—j .      .      .      (44) 

(I       IQ 

in  which  vc  is  the  velocity  of  combustion  under  atmospheric 
pressure  and  10  one-half  the  web  thickness.  From  (12),  Chap- 
ter IV,  we  have 

1/10=  KX0 

which,  when  the  web  thickness  is  burned  through,  gives  for  this 
critical  point, 

KX'0  =  i 

Therefore  from  (44) 

d*l0 

i)c  v7  a  w  <i 
which  gives  X'0  when  vc  and  10  are  known.     Also 

d2l0 

z>c  =  [1.43994]  — — -==       .      .      .     (46) 
A  0  v  dWu 

and 

/0=  [8.56006 -I0]^^    •      •     (460 

While  the  slivers  are  burning  we  have  from  (49) ,  Chapter  IV, 

K 


in  which  X0  refers  to  any  travel  between  u'  and  um  and 

K  =  i  -  (i  -  k)t 
If  we  know  the  values  of  X0  and  k  for  any  point,  we  can 


150  INTERIOR  BALLISTICS 

determine  Nm  by  the  equation 


Therefore  at  the  point  of  breaking  up  into  slivers  (48)  be- 

comes 

~Kf  T^f 

Nm  =  ~r  or  X'0=  --     ....     (49) 


.   0 


The  fraction  kf  which  enters  into  K'  can  be  computed  for  a 
grain  of  given  dimensions  by  (21),  Chapter  III;  and  log  K'  can 
be  taken  from  Table  II  with  k'  as  the  argument.  Therefore 
when  Nm  is  known  Xf0  can  be  found  from  (49),  and  then  x'  ',  taken 
from  Table  I  with  X'0  as  the  argument,  locates  the  point  where 
the  grains  become  slivers  by  the  equation 

u'  =x'  z0      .....     (50) 

In  order  to  determine  Nm  it  is  necessary  to  assume  a  value 
for  km,  or  the  fraction  of  the  entire  charge  burned  at  the  muzzle, 
and  check  this  assumed  value  by  the  given  maxim  am  (crusher- 
gauge)  pressure.  By  (45),  Chapter  IV,  we  have, 


by  means  of  which  Vl  can  be  determined  from  muzzle  data. 
The  constants  M,  N  and  N'  to  be  used  in  the  velocity  and 
pressure  formulas  from  u  =  o  to  u'  are  given  by  the  formulas 

M  =  °^,    N  =  ±  and  N'  =  ^T     •      •     (S2) 

A  o  A  o  X* 

Finally  the  value  of  Mm  for  the  travel  from  u'  to  um  is  given 
by  the  formulas,  deduced  from  (3), 

Mm=^NmVS  =  4NmVS        .     .     .     (53) 

As  an  example  of  this  method  we  will  take  the  mean  crusher- 
gauge  pressure  and  muzzle  velocity  of  five  shots  fired  March  14, 


APPLICATIONS  151 

1905,  with  the  6-inch  Brown  wire  gun,  by  the  Board  of  Ordnance 
at  Sandy  Hook.  The  gun  had  been  previously  fired  twenty-six 
times  with  charges  varying  in  weight  from  32  Ibs.  to  69  Ibs., 
and  at  this  time  was  very  little  eroded.  The  gun  data  are  as 
follows  : 

Vc=  3120  c.  i. 
d  =  6  inches 
um  =  252.5  inches  (total  travel  in  bore). 

The  firing  charge  for  these  five  shots  was  70  Ibs.  of  nitro- 
cellulose powder,  with  8  ounces  of  black  rifle  powder  at  each 
end  of  the  cartridge  for  a  primer.  As  it  is  impossible  to  isolate 
the  action  of  each  kind  of  powder,  we  will  consider  the  charge 
in  its  entirety  and  take  w  =  71  Ibs.*  The  projectiles  varied 
slightly  in  weight  from  100  Ibs.  (about  one-quarter  of  one  per 
cent.)  ;  but  no  material  error  will  result  if  we  make  w  =  100  Ibs. 
The  mean  muzzle  velocity  (vm)  was  3330.4  f.  s.,  and  the  mean 
crusher-gauge  pressure  (pm)  was  42497  Ibs.  per  in.2  The 
charges  were  made  up  of  m.p.  grains  designed  for  an  8-inch 
rifle,  and  of  the  following  dimensions:  R  =  0^.256;  r  = 
o".o255;  m  =  i".c»29.  And,  therefore,  10=  0^.044875;  a  — 
0.72667;  X  =  0.19590;  JJL  =  0.02378;  kf  =  0.85174. 

The  granulation  of  this  powder  is  89  grains  to  the  pound. 
The  volume  of  a  single  grain  computed  by  (15),  Chapter  III,  is 
0.197144  c.  i.;  whence  by  (23'),  Chapter  III,  d  =  1.5776.  From 
these  data  are  found  by  methods  already  fully  illustrated, 

A  =  0.6299 
log  a  =  9.97940  —  10 
Iogz0=  1.82144    .'.  z0=  66.289  in. 


\ogXom=  0.74029 


*  Gossot  recommends  to  increase  the  weight  of  charge  by  one-third  that 
of  the  igniter.  But  there  is  no  practical  difference  in  the  results  by  the  two 
methods. 


152  INTERIOR    BALLISTICS 

We   will   assume    km  =  0.973.      Therefore   from    Table   II, 
\ogKm  =  9.92204  -  10,  and  log  Kf  =  9.78885  -  10.   From  (48) 

we  have 

Km          E!  ... 

N™  =  7JT  =  Txr (54) 

*  -^~om          *•  ^x  o 

whence 

vf       K  Xom  ,     . 

.'.   \QgX'0  =  0.60710 

We  now  find  from  the  preceding  formulas, 

log  VS=  7.44669 
log  M  =  6.70093 
log  M '  =  4.69894 
log  N  =  8.68493  ~~  I0 
logA7/=  7.16201  —  10 

Substituting  these  values  of  M',  N  and  N'  in  equation  (51), 
Chapter  IV,  gives  pm=  42521  Ibs.  per  in.2,  differing  insensibly 
from  the  mean  crusher-gauge   pressure.     The   assumed  value 
of  km  is  therefore  correct.    We  now  find  from  (54)  and  (53), 
\ogNm  =  8.88072  -  10 
logMw=  6.92947 
The  value  of  x'  taken  from  Table  i,  by  means  of  log  X'0,  is 

x'=  1.757. 

Therefore  u'  =  1.757  X  66.289  =  116.47  inches. 
The  two  sets  of  equations  for  velocity  and  pressure  for  a 
charge  of  70  Ibs.,  and  primer  of  i  pound,  are:— 

From  u  =  o  to  u'  =  116.47  inches: 

v*=  [6.70093]  Xt  {i  +  [8.68493  -  10]  X0-  [7.16201  -  io]X2t 
p  =  [4.69894]  X3  {i  +  [8.68493  -  10]  X*-  [7.16201  --  10]  Xb 
From  u'  =  116.47  in.  to  muzzle:— 

&=  [6.92947]  X,  {i  -  [8.88072  -  10]  X0} 
p  =  [4-92748]  X,  {i  -  [8.88072  -  10]  X4} 


} 


APPLICATIONS 


P,  1000.J/ 
75000 


70000- 


v,V 


60000- 


-3000 


55000- 


-2750 


50000- 


10000- 


30000- 


1^000- 


SWW 


-250 


FIG.  4. 


154  INTERIOR   BALLISTICS 

The  X  functions  for  the  travel  u'  are 

log  X'0=  0.60710 
log  X\  =  0.06473 

log X'2  =  9-45763  ~  10 

logJT3  =  9-79332  -  10 

log  ^'4=  0.76498 
logX'5  =  1-48763 

Both  formulas  for  velocity  give  the  same  velocity  for  the 
travel  u',  namely  vf  =  2614  f.  s.  The  pressure  at  this  point  by 
the  first  formula  is  38431  Ibs.;  and  by  the  second  29324  Ibs. 
per  in.2  The  discontinuity  shown  by  the  two  curves  P  and  p 
(see  Fig.  4),  at  the  travel  u1 ',  where  the  grains  break  up 
into  slivers  is  due  to  the  sudden  diminution  of  the  surface  of 
combustion  of  the  grains  at  this  point,  whereby  the  rate  of 
evolution  of  gas  and  heat  suddenly  falls  and  with  it  also  the 
pressure.  In  this  particular  example  the  initial  burning  surface 
of  each  grain  is  3.2  in.2,  and  goes  on  increasing  until  at  the  point 
of  breaking  up  the  vanishing  surface  is  4.2  in.2  It  then  suddenly 
falls  to  about  1.5  in.2,  which  is  approximately  the  surface  of  the 
twelve  slivers.  Of  course  there  is  no  such  absolutely  abrupt 
fall  in  the  pressure  as  is  indicated  by  the  two  pressure  formulas. 
Neither  can  it  be  supposed  that  all  the  grains  maintain  their 
original  form  until  the  web  thickness  is  completely  burned. 
Nevertheless  the  two  pressure  formulas  give  very  approximately 
the  average  pressure  at  or  near  this  point.  It  might  be  possible 
to  connect  the  two  pressure  curves  by  another  curve  of  very 
steep  descent;  but  this  is  hardly  necessary. 

The  characteristics  /  and  vc  are  /  =  1418  and  vc  =  0.134. 
These  characteristics,  computed  with  the  firing  data  of  an  8-inch 
gun,  were  found  on  page  105  to  be  1397  and  0.136,  respectively. 

The  expression  for  y  (powder  burned)  is  by  (45),  Chapter  IV, 

y  =  [4-40457!     - 


APPLICATIONS  155 

For  the  travel  u',  this  formula  gives  y'  =  60.472  Ibs.  =  71  k'. 

At  the  muzzle,  by  the  above  formula,  ym  =  69.085  Ibs.  =  71  km. 

If  it  should  be  found  in  any  case  that  the  powder  was  all 
burned  in  the  gun,  it  would  be  necessary  to  compute  X'0  by  the 
formula 

X'0=K'X,     ......     (56) 

In  this  case  we  should  assume  a  value  for  X0  (or  ~x),  and 
compute  the  maximum  pressure  for  comparison  with  the  crusher- 
gauge  pressure,  following  the  same  steps  as  before. 

We  will  consider  a  few  additional  problems  illustrative  of 
this  method  of  treating  m.p.  grains. 

Problem  i.  —  What  must  be  the  dimensions  of  the  grains  in 
the  example  just  considered  in  order  that  the  combustion  of  the 
entire  charge  may  be  completed  at  the  muzzle?  Also  what 
would  be  the  muzzle  velocity  and  maximum  pressure? 

In  solving  this  problem  we  must  first  consider  the  second 
period  of  combustion,  namely,  that  of  the  slivers.  It  has 
already  been  shown  that  for  a  charge  of  71  Ibs.,  log  VJ  = 
7.44669.  We  also  have  in  this  case,  since  km=  i, 


2X0 

X0  being  the  muzzle  value  of  X0  ;  and  by  (53), 

Mm=4NmVS 
We  thus  find  for  the  second  period  of  combustion, 

v*=  [7.00743]  X,  {  i  -  [8.95868  -  10]  X0)  \A 

P    =    [5.00544]  X,    (l    -    [8.95868   -    10]  X,}  J 

The  muzzle  velocity  by  the  above  equation  is  3376  f.  s.,  an 
increase  of  46  f.  s.,  due  to  the  combustion  of  the  entire  charge 
in  the  gun.  The  muzzle  pressure  is  11433  ^s.  per  in.2 

In  order  to  deduce  equations  for  velocity  and  pressure  for 
the  first  period  of  combustion,  it  will  be  necessary  to  determine 


156  INTERIOR    BALLISTICS 

the  value  of  k'  from  which  to  compute  X'0  and  2  10.     Suppose 
we  adopt  grains  for  which  R/r  —  n  and  m/l0  =  30. 

By  the  method  given  in  Chapter  III,  we  find  for  grains  having 

100              48                     4  81 

these  ratios,  a  =  -r-  ,   X  =  —  ,  M  = and  k  =  — •  .    For 

285 '  199  199  95 

this  value  of  k'  we  find  from  Table  II,  log  K'  =  9.78966;  and 
since  Km  =  i,  we  have  from  (55), 

X'0  =  Kf  Xom 
which  gives 

\ogX'0  =  0.52995. 

By  interpolation  from  Table  i,  we  find  xr  =  1.15217,  and 
then  log  X\  =  9.88302  —  10,  log  X'3  =  9.83966  —  io>  log  X\  = 
0.67980  and  \ogX'6  =  1.32095.  Next  by  equations  (52),  and 
equation  (61),  Chapter  IV,  we  deduce  the  following  equations  for 
velocity  and  pressure,  which  apply  from  u  =  otow'  =  1.1522  X 
66.289  =  76-38  inches: 

v*=  [6.76075]  ^{1  +  18.85245-10]  *0-  [7.24333 -lo]*3.}  - 
p  =  [4-75876]  X3{  i +  [8.85245-10]  X,-  [7.24333-10]  X.  } 

Both  sets  of  equations,  A  and  B,  give  the  same  value  to  z>', 
namely,  2319  f.  s.,  while  the  pressures  at  u'  by  the  two  equations 
are,  respectively,  51723  and  39560  Ibs.  per  in.2,— a  drop  of  more 
than  12000  Ibs.  when  the  grains  break  up  into  slivers.  The 
maximum  pressure  (taking  x  =  0.8)  is  52428  Ibs.  per  in.2 

The  dimensions  of  the  grains  have  yet  to  be  determined. 
We  have  found  for  this  powder  vc  =  0.134  in.  per  sec.  Sub- 
stituting this  and  the  value  of  log  X'0,  given  above,  in  (46')  gives, 

10=  0.038  in. 
and  then 

r  —  l0/2  =  0.019  m- 
R  =  ii  r  =  0.209  in. 
=  1-1     in. 


APPLICATIONS  157 

A  grain  of  these  dimensions  fulfils  all  the  conditions  of  the 
problem.  These  calculations  show  in  a  striking  manner  the 
great  effect  which  minute  variations  (scarcely  measurable)  in 
the  dimensions  of  m.p.  grains  have  upon  the  maximum  pressure, 
increasing  it  in  this  case  by  10,000  Ibs.  per  in.2 

The  cause  of  this  great  increase  in  the  maximum  pressure 
is  that  the  initial  surface  of  combustion  of  the  charge  of  the 
smaller  grains  is  about  15  per  cent,  greater  than  that  of  the 
original  grains,  as  is  easily  shown  by  equation  (26'),  Chapter 
III. 

Problem  2. — What  must  be  the  dimensions  of  the  grains  of 
a  charge  of  71  Ibs.,  in  order  that  the  burning  of  the  web  may 
be  .completed  at  the  muzzle?  Also  determine  the  circumstances 
of  motion. 

To  solve  this  problem  we  obviously  have  u'  =  um;  and 
therefore  x  =  xm  =  3.8091.  As  all  the  X  functions  relate  to 
the  muzzle  only,  we  may  drop  the  accents.  We  have  from 
Table  i,  log  X0  =  0.74029,  log  Xl  =  0.35048,  logX2  =  9.61018, 
logXz  =  9.653 1 1,  log  X4  =  0.91582,  and  log  Xb  =  1.78077.  Sub- 
stituting the  value  of  log  X0  in  (46),  and  making  use  of  the 
known  value  of  vc,  we  find  that  for  the  new  grains, 

lo  =  0".  06098 
Therefore,  as  in  Problem  i,  r  =  —  =  0^.03049 

R  =  5-5*0=  o"-33539 

m  =  3°  lo^  i"-8294 
k'  =  0.85263  =  km 

Since  the  limiting  velocity  Vi  is  independent  of  the  dimen- 
sions of  the  grains,  we  have  as  before,  log  V?  =  7.44669;  and 
this,  with  the  known  values  of  a,  X  and  /*,  substituted  in  equations 
(52),  gives  M,  N  and  N'.  We  thus  derive  the  following  equa- 
tions for  velocity  and  pressure  for  a  charge  of  71  Ibs.  of  these 
particular  grains. 


158  INTERIOR  BALLISTICS 

v*=  [6.55041]  X,(i  +  [8.64211  --  10]  X0-  [6.82262  -  10]  X02} 
p  =  [4.54842]  Xs  { i  +  [8.64211  -  10]  X*-  [6.82262  -  10]  X5  } 

From  these  formulas  we  get  the  following  information: 

Muzzle  velocity,  3118  f.  s. 

Maximum  pressure,  29897  Ibs.  per  in.2 

Muzzle  pressure,  21014  Ibs.  per  in.2 

Powder  burned  in  gun,  60.5  Ibs.  =  71  k' . 

The  maximum  pressure  is  quite  moderate,  owing  to  the 
thickness  of  web  which  gives  an  initial  surface  of  combustion 
but  71  per  cent,  of  that  of  the  original  grains.  The  pressure  is 
well  sustained  to  the  muzzle,  where  it  would  be  considered 
excessive  for  all  except  wire-wound  guns. 

If  we  suppose  the  length  of  the  grains  to  be  twelve  times 

the  web  thickness  we  should  have  a  =  — 5,  X  =  —^.  ju  =  — r-. 

228'          163'  163' 

k'  =  -7,  and  m  =  0.916  in.     Then,  as  before, 

log  M  =  6.56065 
logM'  =  4.55866 
log  N  =  8.60383  -  10 
log  Nf  =  6.90929  —  10 

These  constants  give — 

Muzzle  velocity,  3124  f.  s. 

Maximum  pressure,  30163  Ibs.  per  in.2 

Muzzle  pressure,  20875  Ibs.  per  in.2 

Powder  burned  in  gun,  60.72  Ibs.  =  71  kf 

The  initial  surface  of  combustion  of  the  shorter  grain  is  about 
2.4  per  cent,  greater  than  that  of  the  longer  grain,  which  fact  is 
shown  in  the  maximum  pressures. 

Problem  3. — Suppose  the  powder  we  have  been  considering 
to  be  moulded  into  cylinders  with  an  axial  perforation. 

If  the  length  of  the  grain  is  50  inches  (approximately  the 
length  of  the  cartridge),  and  the  diameter  of  the  axial  perforation 


APPLICATIONS  159 

one-twentieth  of  an  inch,  what  must  be  the  diameter  of  the 
grain  and  thickness  of  web  in  order  that  a  charge  of  71  Ibs.  may 
all  be  burned  just  as  the  shot  leaves  the  muzzle?  Also  determine 
the  equations  for  velocity  and  pressure. 

We  have  already  found  the  thickness  of  web  satisfying  the 
conditions  of  the  problem  to  be  o".  12 196.  (See  Problem  i.) 
Therefore,  by  means  of  the  formulas  pertaining  to  this  form  of 
grain  given  in  Chapter  III,  we  find  the  diameter  of  the  grains 
to  be  o".294  and 

a  =  1.0024392 

X  =  0.0024333 

fJL    =    O 

Since  log  X0  =  0.74029  and  log  Fi2  =  7.44669,  we  find 

v2=  [6.70746]  Xl  {i  -  [6.64590  -  10]  X0} 
p  =  [4.70547]  X3  { i  -  [6.64590  -  10]  X4 } 

which  are  the  equations  required.  The  muzzle  velocity  and 
maximum  pressure  by  these  formulas  are 

vm=  3376  f.  s. 

pm=  37040  Ibs.  per  in.2 

This  latter,  on  account  of  the  smallness  of  N,  occurs  when 
x  =  0.64.  The  muzzle  pressure  is  22750  Ibs.  per  in.2 

A  comparison  of  these  results  with  those  deduced  in  Problem 
i  shows  the  great  superiority  of  the  uniperforated  grain  over 
the  multiperf orated  grain  so  far  as  maximum  pressure  is  con- 
cerned. The  muzzle  velocity  is  the  same  in  both  cases  since  the 
same  weight  of  powder  was  burned  in  the  gun.  But  the  maxi- 
mum pressure  given  by  the  m.p.  grains  is  more  than  15,000  Ibs. 
greater,  and  the  muzzle  pressure  11,000  Ibs.  less  than  with  the 
u.p.  grains.  For  these  latter  grains  the  pressure  is  remarkably 
well  sustained  from  start  to  finish. 

The  monomial  formulas  for  velocity  and  pressure  for  this 
example  are  easily  found  to  be 


l6o  INTERIOR   BALLISTICS 

^  =  [3-35320]  V% 

and 

p  =  [4-70441]  ^3 

The  first  of  these  gives  the  same  value  for  the  muzzle  velocity 
as  the  complete  formula;  while  the  second  gives  maximum  and 
muzzle  pressures  differing  about  o.i  per  cent,  of  their  former 
values. 

During  the  test- firing  of  the  6-inch  Brown  wire- wound  gun 
at  Sandy  Hook,  shots  were  fired  with  charges  varying  from 
32^  Ibs.  to  75  Ibs.,  thus  enabling  us  to  determine  whether  our 
formulas  have  any  predictive  value.  Unfortunately  the  object 
of  the  firing  was  simply  to  test  the  endurance  of  the  gun  and 
no  special  effort  was  made  to  give  to  the  results  any  scientific 
value.  Many  of  the  recorded  velocities  and  pressures  are 
inconsistent  with  each  other  as  when,  more  than  once,  an  increase 
of  charge  gave  a  diminished  velocity  and  pressure.  Some  of 
the  recorded  muzzle  velocities  are  so  manifestly  wrong  that  they 
cannot  be  used  in  getting  averages.  They  suggest  that  the 
chronograph  velocities  were  not  always  reduced  to  the  muzzle. 

We  will  compute  the  new  values  of  /  due  to  a  change  in  the 
weight  of  charge  by  (88),  Chapter  IV,  taking  £0  =  71  Ibs.,  and 

Jo  =  1418  Ibs.  per  in.2,  and  for  a  six-inch  gun,  n  =  —  .      We 

o 

therefore  have 

/  =  [2.53451]  of 

To  determine  X'0,  we  have 

X'.  =  [1.43994!-™ 

vcV  aw  a 

which,  by  substituting  the  known  values  of  d,  10,  vc  and  w,  reduces  to 

[1.52243] 

A   0=   -=- 

v  a  u 


APPLICATIONS 


161 


Nm  is  given  by  (54),  which  easily  reduces  to  (since  log  K' = 

9.78885  -  10) 

^Vw=  [7.96539  -  10]  \/ a  a,      ....      (a) 

Next  we  have  from  equation  (58),  Chapter  IV,  substituting 
for/  its  value  given  above  and  for  w  its  value,  100  Ibs., 

F,2  =  [4.97834!  tf 
and  lastly  from  (53), 

Jf „- 4  ^«Fi!=  [3.54579]  «}av    ...    (6) 

The  following  table  computed  by  these  formulas  shows  the 
agreement  between  the  observed  and  computed  velocities  for 
a  range  of  charges  between  75  Ibs.  and  33^  Ibs.  The  differences 
in  the  last  column  follow  no  apparent  law  and  are  unimportant. 


dj 

Ibs. 

xm 

logMm 

**"m 

Observed 
Velocity 

Computed 
Velocity 

o.-c. 

75-o 

3-9574 

6.95290 

8.87242-10 

3455 

3477 

—  22 

74-5 

3.9383 

6.95008 

8.87347 

3422 

3459 

-37 

73-5 

3-9005 

6.94436 

8-87557 

3402 

3423 

—  21 

72.5 

3-8635 

6.93849 

8.87764 

3380 

3385 

-  5 

71.0 

3-8091 

6.92947 

8.88072 

3330 

3330 

0 

69.0 

3-7392 

6.91693 

8.88474 

3254 

3257 

—  3 

68.0 

3-7052 

6.91047 

8.88672 

3236 

3220 

16 

59-0 

3.4244 

6.84548 

8.90384 

2879 

2888 

-  9 

49.625 

3.1742 

6.76170 

8.92037 

2484 

2536 

-52 

33-25 

2.8146 

6.55588 

8.94643 

1913 

1896 

17 

The  two  sets  of  equations  for  velocity  and  pressure  for  the 
charge  of  75  Ibs.  are: 

From  u  =  o  to  u'  =  117.43  inches: — 

v2=  [6.72436]  X,  {i  +  [8.67663  -  10]  X0-[7.i454i  -  10]  X02} 
p  =  [4.73M  *8  {i  +  [8.67663  -  10]  X,  -[7.14541  -  10]  X,  } 

From  u1 '  =  117.43  in.  to  muzzle: 

ir=  [6.95290]  Xl]{i  -  [8.87242  -  10]  X0} 
p  =  [4.96750]  X3  { i  -  [8.87242  -  10]  X*} 


ii 


!62  INTERIOR   BALLISTICS 

By  the  first  equation  for  pressure  we  find  pm  =  46509  Ibs.  per 
in.2    And  by  the  second,  muzzle  pressure  =  15375  Ibs.  per  in.2 
Both  expressions  for  velocity  give  vf  =  2744  f.  s. 
For  a  charge  of  62  Ibs.,  the  two  sets  of  equations  are 

From  u  =  o  to  u'  =  H4-54  inches:— 

tf=  [6.63999]  Xt  {i  +  [8.70249  -  10}  X0-  [ 
p  =  [4.60288]  X3  {i  +  [8.70249  -  10]  X4-  [ 

From  u'  to  muzzle: 

v2=  [6.86853]  Xi  (i  -  [8.89828  -  10]  X0} 
p  =  [4.83142]  Xs  { i  -  [8.89828  -  10]  X4} 

These  formulas  give  a  muzzle  velocity  of  3,000  f.  s.,  with  a 
maximum  pressure  of  34,263  Ibs.,  and  a  muzzle  pressure  of 
11,784  Ibs.  per  in.2  It  would  seem  as  if  these  last  results  are 
all  that  could  be  desired  for  a  6-inch  gun. 

APPLICATION  TO  THE  FOURTEEN-!NCH  RIFLE 

The  i4-inch  rifle  was  designed  by  the  Ordnance  Department 
to  give  a  "muzzle  velocity  of  2,150  f.  s..to  a  projectile  weighing 
i, 660  Ibs.,  with  a  charge  of  nitrocellulose  powder  of  about  312 
Ibs.,  and  with  a  maximum  pressure  not  to  exceed  38,000  Ibs.  per 
square  inch."  The  gun  has  a  powder-chamber  capacity  of 
13,526  cubic  inches  and  a  travel  of  projectile  in  the  bore  of 
413.85  inches.  The  type  gun  has  been  fired  to  date  55  times 
with  charges  varying  from  102^  to  328  Ibs.,  producing  muzzle 
velocities  ranging  from  901  to  2,252  f.  s.,  and  crusher-gauge 
pressures  from  4,875  to  46,078  Ibs.  per  in.2,  this  latter  with  a 
charge  of  326  Ibs. 

The  powder  employed  was  "  International  Smokeless  powder, 
lot  i,  1906,  for  1 2 -inch  gun."  The  grains  were  cylindrical 
multiperf orated  (7  perforations),  of  the  following  dimensions: 


APPLICATIONS  163 

Outside  diameter,  0.826  in. 
Diameter  of  perforations,  0.0815  in. 
Length,  1.883  m- 
Thickness  of  web,  0.145375  in. 
These  dimensions  give: 

a  =  0.71584 

X  =  0.20974 

/A  =  0.02151 

k'  =  0.85058 

log^'  =  9.78778-10. 

The  granulation  of  the  powder  is  20.6  grains  to  the  pound, 
which  by  (24'),  Chapter  III,  makes  the  density  (5)  1.4291. 

We  will  base  our  calculations  on  round  No.  55,  fired  January 
23,  1911,  with  a  charge  of  328  Ibs.  of  nitrocellulose  powder  plus 
an  "igniter"  of  9  Ibs.  of  rifle,  or  saluting,  powder.  This  round 
affords  the  following  data: 

co  =  337  Ibs. 

w  =  1664  Ibs. 

vm  =  2252  f.s. 

pm  =  43640  Ibs.  per  in.2 

The  preliminary  calculations  give 

A  =  0.68965 
log  a  =  9.87523-10 
Iogz0  =  1.65768 

xm  =  9.1025 
log  Xom  =  0.87855 
log  Xim  =  0.60885 

By  a  few  trials  it  will  be  found  that  the  observed  values  of 
vm  and  pm  are  satisfied  when  km  =  0.953  and  therefore  from 
Table  II,  log  Km  =  9.89388-10.  We  also  find  log  X'0  =  0.77245, 
log  V,2  =  6.99575,  xr  =  4-6354  and  u'  =  210.75  inches. 


164  INTERIOR   BALLISTICS 

The  equations  of  the  velocity  and  pressure  curves  are  found 
to  be 

Fromw  =  otow'  =  210.75  in.: 

v2  =  [6.07812]  Xi  {i  +  [8.54923-10]  Xo  ~  [6.78775-10]  X02} 
p  =  [4-72511]  Xs{i  +  [8.54923-10]  X,  -  [6.78775-10]  Xb} 

From  uf  =  210.75  in.  to  muzzle: 

v2  =  [6.31211]  Xl  {i  -  [8.71430-10]  X0] 
p  =  [4.959*0]  Xs{i  -  [8.71430-10]  X*\ 

Both  of  the  velocity  formulas  give  v'  =  1921  f.  s.  The  first 
formula  for  pressure  gives  p'  =  27457  and  the  second  19,772 
Ibs.  per  in.2  The  muzzle  pressure  comes  out  9,485  Ibs.  per  in.2 

This  round  makes  the  powder  characteristics,  by  (64)  and 
(67),  Chapter  IV, 

/  =  1759-7 
vc  =  0.10214 

For  computing  the  velocity  and  pressure  constants  when  the 
charge  varies,  we  will  consider  vc  constant  and  assume  /  to  vary 
directly  as  the  weight  of  charge.  That  is,  we  will  compute  /  by 
the  formula 


Equation  (69),  Chapter  IV,  becomes,  by  substituting  the 
values  of  J2,  10  and  vc, 

[3.58446] 

......      (b) 


Also  (58),  Chapter  IV,  becomes,  by  employing  the  expression 
for  /given  above, 


\ 


APPLICATIONS 


We  then  have 


ad>  \? 


-  fa&  \ 
<°  VT^/ 


GO 


and 


~X 


N'  =  -~  N*  =  [9.68929-10] 

A"" 


By  (49),  we  have 


K' 


Combining  this  with  the  expression  for  N  we  have 
Nm  =  -~  =  (0.16507)  N      .      .      . 
Finally  we  have  from  (53) 

Mm  =  4  Nm  Ff  =  —  =  (0.23399)  M 


(g) 


(K) 


The  following  table  gives  the  computed  muzzle  velocities 
and  maximum  pressures  for  certain  charges,  computed  by  these 
formulas,  together  with  the  observed  velocities  and  crusher- 
gauge  pressures  for  comparison: 


M 

Iha 

w 
It.- 

Observed 
Velocity, 

Computed 
Velocity, 

O.-C. 

f    c 

Observed 
Pressure, 

Computed 
Pressure, 

O.-C. 

f.s. 

f.s. 

Ibs.  per  in.'J 

Ibs.  per  in.2 

337 

1664 

2252 

2252 

0 

43640 

43628 

12 

335 

1660 

2238 

2240 

—    2 

42811 

42944 

-133 

334 

1660 

2232 

2232 

0 

42877 

42637 

240 

284 

1662  yz 

1857 

1871 

-14 

25530 

29142 

—  3612 

263 

1660 

1738 

1724 

14 

21190 

24431 

-3241 

239 

1660 

1567 

1556 

II 

16795 

19704 

-2909 

The  greatest  difference  between  the  observed  and  computed 
muzzle  velocities  is  considerably  less  than  one  per  cent,  and 


1 66  INTERIOR   BALLISTICS 

may  be  disregarded.  The  same  is  true  of  the  differences  of  the 
observed  and  computed  maximum  pressures  of  the  first  three 
charges.  Then,  as  the  charges  are  greatly  reduced,  these  differ- 
ences are  largely  increased.  This  may  be  accounted  for  if  the 
same  kind  of  copper  cylinders  were  employed  for  all  the  charges. 
For  a  charge  of  314  Ibs.  of  service  powder  and  an  igniter  of 
9  Ibs.  of  rifle  powder,  making  o>  =  323  Ibs.,  and  density  of  loading 
0.66 1,  the  equations  for  velocity  and  pressure  are  as  follows: 

From  u  =  o  to  u'  =  208.75  inches. 

ir  =  [6.05003]  Xi!i  +  [8.55696-10]  x0  -[6.80321  -- 10]  x2.} 

p  =  [4-67944]  X3  [i  +  [8.55696-10]  X<  -  [6.80321  --  10]  X,} 

From  u'  =  208.75  inches  to  muzzle. 

v  =  [6.28402]  X,  {i  -  [8.72203-10]  X0} 
p  =  [4.91343]  ^3  {i  ~  [8.72203-10]  X4} 

These  formulas  give 

/  -  1686.6 

Muzzle  velocity  =  2152  f.  s. 
Maximum  pressure  =  39351  Ibs.  per  in.2 

This  muzzle  velocity  is  that  for  which  the  gun  was  designed, 
but  the  maximum  pressure  is  about  3^  per  cent,  greater.  The 
muzzle  pressure  comes  out  8714  Ibs.  per  in.2 

Example. — Suppose  the  volume  of  the  powder  chamber  to 
be  increased  (as  is  proposed  by  the  Ordnance  Department)  to 
15,000  cubic  inches,  by  lengthening  the  chamber  6.65  inches, 
thereby  reducing  the  travel  of  the  projectile  to  407.2  inches. 
If  the  density  of  loading  remain  0.66 1,  what  would  be  the 
charge,  the  muzzle  velocity,  and  maximum  pressure? 
Answers : 

£  =  349-2  +  9  =  358.2  Ibs. 
pm  =  41683  Ibs.  per  in.2 
M .  V.    =  2233.5  f-  s. 


APPLICATIONS  167 

With  a  charge  of  337  Ibs.  of  service  powder  and  an  igniter 
of  9  Ibs.  of  black  powder,  we  should  get,  with  the  lengthened 
chamber,  a  muzzle  velocity  of  2150  f.  s.,  with  a  maximum 
pressure  of  about  38,400  Ibs.  per  in.2  These  results  are  prac- 
tically those  sought  for  in  designing  the  present  1  4-inch  gun. 

Example  2.  —  Suppose,  instead  of  enlarging  the  powder 
chamber  of  the  1  4-inch  gun,  we  lengthen  the  grains  of  powder, 
and  employ  the  ratios  R/r  =  n  and  m/l0  =  200.  These  ratios 
give,  as  is  shown  in  Chapter  III, 

1210 

a  =  -      -  =  0.64158 
1900 

=  0.3l829 


1219 
fJL   = 


—  =    —    0.00328 


1219 

.  0.84368 


1900 

logtf'  =  9.78150  -  10.     (By  Table  II.) 

Employing  these  grains,  what  muzzle  velocity  and  maximum 
pressure  may  be  expected  with  a  charge  of  314  Ibs.  of  service 
powder  and  an  igniter  of  9  Ibs.  of  black  powder,  in  the  gun  as 
it  is  now,  where  Vc  =  13526  c.  i.,  and  um  =  413.85  in.  ? 

The  preliminary  calculations  give: 

A  =  0.661 
log  a  =  9.91017 
Iogz0  =  1.67419 
xm  =  8.7630 
log  Xom  =  0.87273 
log  Xlm  =  0.59873 

By  equations  (a)  to  (ti),  inclusive,  we  find,  the  web  thickness 
remaining  as  before, 


1  68  INTERIOR  BALLISTICS 

/  =  1686.6 

log  Fi2  =  6.95993 

logXf0  =  0.76472     .'.  x'  =  4-4201  and  u'  =  208.75  in- 

The  equations  for  velocity  and  pressure  are 

From  u  =  o  to  u'  =  208.75  m-: 

v2  =  6.00247  X,  {  i  +  [8.73812-10]  X0  -  [5.98664  -  10]  X02 
p  =  4.63188  X3  {  i  +  [8.73812-10]  X,  -  [5.98664  -  10]  X6 

From  u'  =  208.75  in.  to  muzzle: 

v2  =  [6.27775]  Xl  {i  -  [8.71576  -  10]  X0} 
p  =  [4.90716]  *i'fi  -  [8.71576  ~  10]  X,} 
From  these  equations  we  find, 

Maximum  pressure  =  37851  Ibs.  per  in.2 
Muzzle  velocity  =  2146  f.  s. 
Muzzle  pressure  =  8789  Ibs.  per  in.2 

v'  =  1820.3  f.  s. 

.  ,  26299 


The  dimensions  of  these  grains  are  found  from  the  ratios 
given  above,  and  are  as  follows: 

Diameter  of  perforations  =  10  =  0.0727  in. 
Diameter  of  grain  =  1  1  10  =  0.8  in. 
Length  of  grain  =  200  10  =  14.54  in. 

The  following  table  gives  the  pressures  (p')  at  different  points 
of  the  bore  for  a  charge  of  314  Ibs.  of  service  powder  plus  an 
igniter  of  9  Ibs.,  making  a>  =  123  Ibs.,  and  also  the  pressures 
(p")  of  the  same  charge  made  up  of  the  grains  whose  dimen- 
sions are  given  above. 

It  will  be  seen  from  this  table,  and  the  previous  calculations, 
that  increasing  the  length  of  the  powder  grains  relieves  the 
maximum  pressure  more  than  is  accomplished  by  lengthening 
the  powder  chamber,  for  the  same  muzzle  energy: 


APPLICATIONS 


169 


X 

u 
Inches. 

P' 

Ibs.  per  in.2 

P" 

bs.  per  in.2 

P'-P" 

Remarks. 

0.  I 

4.72 

24325 

22388 

1937 

0.2 

9-45 

31329 

29138 

2191 

0-3 

14.17 

35084 

32888 

2196 

0.4 

18.89 

37234 

35I3I 

2103 

05 

23.61 

38453 

36483 

1970 

0.6 

28.34 

39090 

37275 

1815 

0.7 

33-o6 

39351 

37691 

1660 

0.8 

37-78 

39351 

37851 

1500 

Maximum  pressure. 

0.9 

42.50 

39I8I 

37833 

1348 

I.O     . 

47-23 

38892 

37690 

1202 

i-5 

70.84 

36650 

36065 

585 

2.0 

94  45 

34150 

34019 

131 

2-5 

118.07 

31849 

32060 

—    211 

3-0 

141.68 

29820 

30293 

-  473 

3-5 

165.29 

28049 

28726 

-  677 

4.0 

188.91 

26500 

27341 

—  841 

4.4201 

208.75 

25344 

26299 

-  955 

The  web  thickness  is 

5-0 

236.13 

16308 

16295 

+     13 

burned    at    this 

point. 

6.0 

283.36 

13566 

13585 

-     19 

7.0 

330-59 

H45i 

11496 

—    45 

8.0 

377.82 

9775 

9838 

-     63 

8-763 

4I3-85 

8714 

8789 

-     75 

Muzzle. 

CHAPTER    VI 

ON    THE  RIFLING    OF    CANNON 

Advantages  of  Rifling. — The  greater  efficiency  of  oblong 
over  spherical  projectiles  is  twofold.  In  the  first  place  they 
have  greater  ballistic  efficiency, — that  is,  for  the  same  caliber, 
muzzle  velocity  and  range,  an  oblong  projectile  has  a  higher 
average  velocity  during  its  flight  than  a  spherical  projectile. 
This  gives  to  the  former  a  flatter  trajectory  which  increases 
the  probability  of  hitting  the  target.  Experimental  firing  has 
demonstrated  that  the  mean  deviation  of  the  shots  from  a  rifled 
gun  at  medium  ranges,  when  all  known  and  controllable  causes 
of  deviation  have  been  eliminated,  is  only  one-third  that  from 
a  smooth  bore.  This  advantage  results  both  from  the  greater 
sectional  density  of  the  oblong  projectile  whereby  it  is  enabled 
the  better  to  overcome  the  resistance  of  the  air,  and  also  because 
this  resistance  is  diminished  by  the  more  pointed  head. 

In  the  second  place  the  penetration  of  oblong  projectiles, 
other  things  being  equal,  is  much  greater  than  can  be  realized 
with  spherical  shot,  while  the  bursting  charge  of  oblong  shells  is 
as  great  or  even  greater  than  that  of  spherical  shells  on  account 
of  their  greater  length.  These  are  very  substantial  advantages; 
but  to  secure  them  it  is  essential  that  the  oblong  projectile  should 
keep  point  foremost  in  its  flight,  otherwise  it  would  have  neither 
range,  accuracy  nor  penetration,  but  would  waste  its  energy 
beating  the  air. 

The  only  way  to  secure  steadiness  of  flight  to  an  oblong 
projectile  is  to  keep  its  geometrical  axis  in  the  tangent  to  the 
trajectory  it  describes  by  giving  it  a  high  rotary  velocity  about 
this  axis.  This  is  accomplished  by  rifling,  as  it  is  called, — that 

170 


ON   THE   RIFLING   OF   CANNON  iyi 

is,  by  cutting  spiral  grooves  in  the  surface  of  the  bore  into  which 
a  projecting  copper  band,  securely  encircling  the  projectile 
near  its  base,  is  forced  as  soon  as  motion  of  translation  begins, 
thus  giving  to  the  projectile  a  rotary  motion  in  addition  to  its 
translation  as  it  moves  down  the  bore.  The  rifling  may  be 
such  that  the  grooves  (or  rifles)  have  a  constant  pitch,  that  is, 
make  a  constant  angle  with  the  axis  of  the  bore;  or,  this  angle 
may  increase.  In  the  first  case  the  gun  is  said  to  be  rifled  with 
a  constant  twist,  and  in  the  second  case  with  an  increasing 
twist.  In  all  cases  the  twist  at  any  point  of  the  bore  is  measured 
by  the  linear  distance  the  projectile  would  advance  while  making 
one  revolution  supposing  the  twist  at  that  point  to  remain 
constant.  This  linear  distance  is  always  expressed  in  calibers, 
and  is  therefore  independent  of  the  unit  of  length  employed. 

The  Developed  Groove.  Uniform  Twist. — The  element  of  a 
groove  of  uniform  twist  developed  upon  a  plane  is  evidently 
a  right  line  A  C  making,  with  the  longitudinal  element  of  the 
surface  of  the  bore  A  B,  the  constant  angle  B  A  C,  whose 
tangent  is  B  C/A  B*  Suppose  A  B  to  be  the  longitudinal  ele- 
ment passing  through  the  beginning  of  the  groove  at  A,  which  is 
near  the  base  of  the  projectile  when  in  its  firing  seat  and  directly 
in  front  of  the  rotating  band.  Make  A  B  =  nd,  n  being  the 
number  of  calibers  the  projectile  travels  while  making  one 
revolution.  Then  B  C  will  be  equal  to  the  circumference  of 
the  projectile;  or,  B  C  =  TT  d.  If  we  designate  the  angle  of 
inclination  of  the  groove,  B  A  C  by  /?,  we  shall  have 

BC       ird        TT 

tan  )8  =  -7-5  =  — i  =  -       .      .      .      .     (i) 
A  B      nd       n 

Increasing  Twist. — With  a  uniform  twist  the  maximum 
pressure  produced  on  the  lands  (or  sides  of  the  grooves)  occurs 
(as  will  be  shown  presently)  at  the  point  of  maximum  pressure 

*  The  simple  diagrams  required  in  this  Chapter  can  easily  be  constructed 
by  the  reader. 


172 


INTERIOR   BALLISTICS 


on  the  base  of  the  projectile,  which  point,  as  we  know,  is  near  the 
beginning  of  motion.  From  this  point  the  pressure  on  the*  lands 
decreases  to  the  muzzle  where  it  is  not  generally  more  than  one- 
fourth  of  its  maximum  value.  It  is  considered  by  gun-designers 
a  desideratum  to  have  the  pressure  on  the  lands  as  uniform  as 
possible,  and  to  this  end  recourse  is  had  to  an  increasing  twist- 
that  is,  the  angle  which  a  groove  makes  with  the  axis  of  the  bore, 
instead  of  being  constant  as  with  a  uniform  twist,  increases  from 
the  beginning  of  rifling  toward  the  muzzle.  If  this  variable  angle 
be  represented  by  0,  we  shall  still  have,  as  before, 

tan  0  =  - (i) 

in  which  n  is  now  a  variable,  decreasing  in  value  as  the  distance 
from  the  origin  of  rifling  increases.  If,  as  before,  we  take  the 
origin  of  rectangular  coordinates  at  A,  the  beginning  of  rifling, 
and  suppose  A  B  to  be  a  longitudinal  element  of  the  bore  and 
B  C  the  length  of  arc  revolved  through  by  a  point  on  the  surface 
of  the  projectile  while  it  travels  from  A  to  B,  then  the  developed 
groove,  A  C,  is  a  curve  convex  toward  A  B,  the  axis  of  abscissas, 
for  the  reason  that  by  definition  tan  0  increases  from  A  toward 
B.  The  two  forms  of  increasing  twist  that  have  been  most 
generally  adopted  are  the  parabolic  and  circular. 

General  Expression  for  Pressure  on  the  Lands. — Before 
attempting  to  decide  upon  the  best  system  of  rifling  it  will  be 
necessary  to  deduce  an  expression  for  the  pressure  upon  the 
lands.  Take  a  cross-section  of  the  bore  and  suppose  for  sim- 
plicity that  there  are  only  two  grooves  opposite  to  each  other, 
and  let  the  prolongation  of  the  bearing  surface  pass  through  the 
axis  of  the  bore,  as  is  practically  realized  in  the  latest  systems. 
Let  M  represent  the  point  of  application  of  the  bearing  surface 
of  the  upper  groove.  We  will  take  three  coordinate  axes: 
one  axis  (x)  is  coincident  with  the  axis  of  the  bore,  while  the  others 
(y  and  z)  are  in  the  plane  of  the  cross-section  of  the  bore  and 


ON   THE    RIFLING    OF    CANNON  173 

perpendicular  to  each  other.  Let  the  axis  y  be  directed  along 
O  M,  and  z  in  a  direction  perpendicular  to  0  M. 

The  pressure  at  M ,  whatever  may  be  its  direction,  can  be 
replaced  by  the  three  following  mutually  perpendicular  com- 
ponents: The  first,  perpendicular  to  the  axis  of  the  bore  and 
consequently  acting  along  the  radius  through  M;  the  second 
lying  in  the  plane  tangent  to  the  surface  of  the  bore  (normal 
to  the  radius  O  M)  and  acting  along  the  normal  to  the 
groove;  the  third  lying  in  this  same  plane  and  tangent  to  the 
groove. 

The  first  of  these  components  will  be  destroyed  by  the 
similar  component  of  the  opposite  groove  and  does  not  enter 
into  the  equation  of  motion  of  the  projectile.  The  second 
component,  which  is  the  normal  pressure  against  the  bearing 
surface  of  the  groove,  we  will  designate  by  R.  The  third 
component,  being  in  the  tangent  to  the  groove,  represents  the 
friction  on  the  guiding  side  of  the  groove,  and  may  be  designated 
by  f  R,  in  which  /  is  the  coefficient  of  friction.  The  forces  R 
and  /  R  give  the  following  components  along  the  axes  x  and  z: 

Axis  of  x.  Axis  of  z. 

Force  R   .      .     -  R  sin  0  7?  cos  0 

Force  fR     .     -  f  R  cos  0         -fRsm0 

The  positive  direction  of  the  axis  of  x  is  toward  the  muzzle; 
that  of  z  in  the  direction  of  the  force  R,  and  0  is  the  angle  which 
the  groove  makes  with  the  axis  of  x.  The  full  component  for 
the  upper  groove  is : 

On  axis  of  x  .      .      -  R  (sin  0  +  /  R  cos  0) 
On  axis  of  z     .  R  (cos  0  —  f  R  sin  0) 

For  the  lower  groove  the  component  along  the  axis  of  x  has 
the  same  value  and  sign  as  the  upper  one;  while  the  component 
along  the  axis  of  z  has  the  same  value  but  the  opposite  sign. 
Besides  these  forces  the  projectile  is  also  subjected  to  the  variable 


174  INTERIOR   BALLISTICS 

pressure  of  the  powder  gases  on  its  base  acting  along  the  axis  of  x 
in  the  positive  direction,  which  force  call  P.  If  we  replace  the 
grooves  by  the  forces  enumerated  above,  we  may  consider  the 
projectile  a  free  body  and  apply  to  it  Euler's  equations.  These 
equations  are  six  in  number;  but,  as  is  readily  seen,  they  reduce 
to  two  in  the  problem  under  consideration,  namely:  an  equation 
of  translation  along  the  axis  of  x,  and  of  rotation  about  the  same 
axis,  or,  what  is  the  same  thing,  the  axis  of  the  projectile.  The 
first  equation  is 

M-^=  P  -  2  R  (sin  6  +  /cos0)     ...     (2) 
and  the  second 

-fsmd)      ...     (3) 


in  which  r  is  the  radius  of  the  projectile,  co  its  angular  velocity 
about  its  axis  and  k  its  radius  of  gyration. 

The  angular  velocity  co  of  a  projectile  about  its  geometrical 
axis  for  an  increasing  twist,  continually  increases  as  it  moves 
along  the  bore  from  zero  to  its  muzzle  value,  which  is  IT  v/n  r, 
v  being  the  muzzle  velocity  of  translation  and  r  the  radius  of 
the  projectile.  Its  magnitude  at  any  instant  is  given  by  the 
equation 


where  <p  is  the  angle  turned  through  from  the  beginning  of 
motion  expressed  in  radians.  The  angular  acceleration  is 
found  by  differentiating  this  equation  with  respect  to  the  time, 
which  gives 

A  d  u       d?(p 

Angular  acceleration  =  -j-  =  -r-      ...     (4) 

If  we  now  take  x  and  y  as  the  rectangular  coordinates  of  the 
developed  groove  with  the  origin  at  the  beginning  of  rifling  and 
the  axis  of  abscissas  parallel  to  the  axis  of  the  bore,  then  x  will 


ON   THE   RIFLING    OF   CANNON 


175 


represent  at  any  instant  the  distance  travelled  by  the  shot,  and 
the  corresponding  value  of  y  will  be  r<p,  where  r  is  the  radius  of 
the  projectile.  Substituting  y  for  <p  in  (4),  gives 

da}  _  i  d2y 

~dt~"~rdP  .......     (5' 

Substituting  this  value  of  d  u/d  t  in  (3),  it  becomes,  putting 


M  fji-j     =  2R  (cos  6  -  /sin  d)    .      .      .      .     (6) 

Before  these  equations  can  be  used  for  determining  2  R  we 
must  eliminate  d  t;  and  this  we  can  do  by  means  of  the  equation 
of  the  developed  groove.  Let 

y=f(x) 
be  this  equation.    Then  employing  the  usual  notation  we  have 

j|  -/'(*)»  tan  «    .....     (7) 
and 

&-*••<* 

Also,  since 

dy  _  d  y    dx  doc 

~ii^Tx'~dl  =  f^~dt' 
we  have,  by  differentiating, 


d2  y  d2  x 

Substituting  this  value  of  -7—  ,    and  also  the  value  of  -r^- 

from  (2)  in  (6),  and  solving  for  2  R,  we  have 


2  R  =  (  , 

'  i  -tan  0{/-  M  (/+  tan0)}  ' 

In  using  this  equation  /"  (x)  and  tan  6  are  obtained  from  the 
equation  of  the  developed  groove,  M  and  M  from  the  projectile, 


INTERIOR   BALLISTICS 

V  and  P  from  the  equations  for  velocity  and  pressure  deduced 
in  Chapter  IV,  while  /  is  determined  by  experiment.  The 
resulting  value  of  2  R  will  be  sum  of  the  rotation  pressures  on 
all  the  lands. 

Uniform  Twist. — For  uniform  twist  f'(x)  is  zero  and  0 
becomes  constant  and  equal  to  0.  Its  value  is  given  by  (i). 
Making  these  substitutions  the  expression  for  2  R  becomes 
for  uniform  twist 

IJL  P  tan  0  sec  0 ,  ^ 

2  R  =i  -/tan  0  +  M  tan  ft  (f  -ftan  0) 

In  the  second  member  of  (9),  P,  the  pressure  on  the  base  of 
the  projectile,  is  the  only  variable;  and  therefore  2  R  is  directly 
proportional  to  this  pressure,  and  is  a  maximum  when  P  is  a 
maximum.  In  equation  (8)  there  are  four  variables  in  the 
second  member,  namely,  P,  v,  6  and/"  (x) ;  and  it  is  not  obvious 
on  simple  inspection  where  the  point  of  maximum  rotation  is 
located.  It  will  be  shown,  however,  by  examples  that  for  an 
increasing  twist  this  point  is  much  nearer  the  muzzle  than  when 
the  twist  is  uniform. 

Increasing  Twist.  Semi-Cubical  Parabola. — To  continue 
an  increasing  twist  quite  up  to  the  muzzle  must  conduce  to 
inaccuracy  of  flight,  and  especially  so  when  the  projectile  has 
partially  left  the  bore  so  that  it  has  lost  its  centering.  To 
remedy  this  the  acceleration  of  rotation  near  the  muzzle  is  made 
either  zero  or  constant  (preferably  the  former),  in  order  to 
relieve  the  rotating  band  as  much  as  possible  from  pressure  and 
to  reduce  the  torsional  effect  upon  the  gun,  far  removed  from 
its  support  at  the  trunnions.  In  all  of  our  sea-coast  guns  the 
final  twist  is  made  constant  beginning  at  about  2^2  calibers  from 
the  muzzle.  The  developed  groove  for  the  increasing  twist  is 
a  semi-cubical  parabola  whose  general  equation  is 

y  +  b  =  p  (x  +  rf (10) 


ON   THE   RIFLING    OF   CANNON  177 

The  axis  of  x  is  parallel  to  the  axis  of  the  bore  and  the  origin 
is  at  the  beginning  of  the  rifling,  just  in  front  of  the  rotating 
band  of  the  projectile  when  in  its  firing  seat.  The  coordinates 
of  the  vertex  of  the  parabola  are  -a  and  -b,  and  these  with  the 
parameter  p  are  determined  by  the  particular  twist  adopted 
for  the  beginning  and  ending  of  the  increasing  twist.  Suppose 
the  rifling  to  start  with  a  twist  of  one  turn  in  nv  calibers,  and 
that  at  2}/2  calibers  from  the  muzzle  where  the  rifling  begins 
to  be  constant  it  has  a  twist  of  one  turn  in  n2  calibers  (HI  >  n^). 
For  these  two  points  we  have,  by  (i), 

tan  61  =  -  -  and  tan  02  =  — 
n,  n2 

0!  and  62  being  the  inclinations  of  the  grooves  with  the  axis  of 
x  at  the  points  considered.     Differentiating  (10),  we  have, 

^  =  tan0=|/>(*  +  <*)2         .      .      .     (n) 

At  the  origin  x  =  o,  which  gives 

3  p  \/a        TT 


At  2^2  calibers  from  the  muzzle  where  the  increasing  twist 
ends,  x  —  u2,  and  we  have  at  this  point 


2  n% 

From  these  two  last  equations  we  find 

"— .  («) 


and 


1  78  INTERIOR   BALLISTICS 

Since  at  the  origin  x  and  y  are  both  zero,  we  find  from  (10) 
and  (13), 


2-n-a 


Thus  all  the  constants  in  the  equation  of  the  developed 
groove  are  given  in  terms  of  HI,  n2  and  u2.  Lastly,  differentiating 
(n)  gives 

/'(*)=—  ^L=  .....     (15) 
4  V  x  +  a 

If  the  vertex  of  the  semi-cubical  parabola  is  at  the  origin,  or 
beginning  of  rifling,  a  and  b  are  zero,  and  (10)  becomes 

y  =  P  x*   .......     (16) 

In  this  case  the  twist  is  zero  at  the  origin  and  increases  to 
one  turn  in  n2  calibers  near  the  muzzle.  The  values  of  tan  0, 
p  and  j"  (x)  for  this  particular  form  of  rifling  are  deduced  from 
(n),  (13),  and  (15),  by  making  a  zero.  This  form  of  groove  is 
that  adopted  by  the  navy  for  all  their  heavy  guns  of  recent 
construction. 

Common  parabola.  —  The  equation  of  the  common  parabola  is 

y  +  b  =  p  (x  +  a)2    .....     (17) 

where  —  a  and  —  b  are  the  coordinates  of  the  vertex.  The 
constants  are  determined  as  for  the  semi-cubical  parabola,  and 
are  as  follows: 


2  n 


(19) 

>     Trr 


i 

7T 


t       2n2(u2+a) 

f"  (*}=2P  (21) 


ON    THE    RIFLING    OF    CANNON  179 

Relative  Width  of  Grooves  and  Lands.—  In  our  service 
siege  and  sea-coast  guns  the  number,  N,  of  grooves  (or  lands) 
is  given  by  the  equation 

N  =  6  d 

in  which  d  is  the  diameter  of  the  bore  in  inches,  and  is  a  whole 
number  for  each  of  these  guns.  If  wg  is  the  width  of  a  groove 
and  Wi  the  width  of  a  land,  we  have  the  relation 


inches- 


The  best  authorities  lay  down  the  rule  that  the  width  of  a 
groove  should  be  at  least  double  that  of  a  land.  In  our  guns 
the  lands  are  made  0.15  in.  wide,  and  the  grooves  are  therefore 
0.5236  —  0.15  =  0.3736  in.  wide. 

Application  to  the  lo-inch  B.  L.  R.  Model  1888.—  This  gun 
has  60  grooves  which,  beginning  at  20.1  inches  from  the  bottom 
of  the  bore  with  a  twist  of  one  turn  in  50  calibers,  increase  to 
one  in  25  at  20  inches  from  the  muzzle,  and  from  thence  continue 
uniform.  We  therefore  have  HI  =  50  and  n2  =  25.  The  bore 
is  22.925  ft.  long,  and  therefore  u2  =  19.583  ft.  The  developed 
groove  is  a  semi-cubical  parabola  whose  equation  is  (10).  The 
constants  are 

a  =  19^3  =  6.52g  ft. 


6.S287T 

b  =  —  --  =  0.27344  ft. 

.75 

P  =  "    —=5  =  °-°l6395- 
.528 


The  equation  of  the  developed  groove  (changing  x  to  u  to 
indicate  travel  of  projectile)  is  therefore, 

y  +  0.27344  =  0.16395  (u  +  6.528)*  -     (22) 

in  which  y  will  be  given  in  feet. 


l8o  INTERIOR   BALLISTICS 


From  (n),  we  have 


tan  0  =  0.024592  \/  u  +  6.528 
Making  u  zero  in  this  last  equation  gives 

»i=3°35'4*" 

which  is  the  inclination  at  which  the  groove  starts.  At  20  inches 
from  the  muzzle  where  the  twist  becomes  uniform  (and  which  is 
therefore  a  point  of  discontinuity  on  the  developed  groove)  we 
have  u-i  =  19.583;  and  at  this  point 

*>=7°9'45" 

This  value  of  0  is  retained  to  the  muzzle. 
From  (15),  we  have 

/"  fx\  =     o-oiffg^ 
Vu  +6.528 

This  function   decreases   from   the   origin   to   the  point  of 
discontinuity.     From  this  point  to  the  muzzle  /"  (x)  is  zero. 
If  we  put 

K  =  _  Msec  0 

"  i  -  tan  0{  f  -  n  (f  +  tan  0) }  ' 

equation  (8)  becomes 

2  R  =  K  [P  tan  0  +  M  ir  f"  (x) }       .      .      (24) 

Captain  (now  Sir  Andrew)  Noble,  as  the  result  of  very  careful 
experiments  made  by  him  with  i2-cm.  quick-firing  guns,  found 
/  =  0.2,  and  this  value  will  be  adopted  in  what  follows.  We 
also  have  for  cored  shot  /*  =  0.5,  nearly.  Substituting  these 
values  of  /  and  /*  in  (23),  it  will  be  found  that  K  increases  very 
slowly  as  0  increases.  The  values  of  K  for  u  =  oandw  =  19.583 
are,  respectively,  0.5032  and  0.5064.  We  might  therefore  take 
for  K  the  arithmetical  mean  of  these  two  values  and  write  (24) 

2  R  =  0.5048  { P  tan  0  +  M  v~  f"  (x) }  (25) 


ON    THE    RIFLING    OF    CANNON  l8l 

without  any  material  error.     This  formula  may  be  employed 
for  all  our  sea-coast  guns. 

If  the  lo-inch  gun  were  rifled  with  the  kind  of  groove  given 
by  (16),  we  should  find 

2  7T 

P     =    -  X  -  --    =      O.OI893I 


tan  0  =  0.028397  V  u 


In  this  form  of  rifling  the  initial  inclination  of  the  grooves 
is  zero  and  increases  to  7°  9'  45"  at  20  inches  from  the  muzzle, 
where  the  twist  becomes  uniform.  Between  this  point  and  the 
muzzle,  j"  (x)  is  zero. 

Uniform  Twist. — If  we  suppose  the  lo-inch  gun  to  be  rifled 
throughout  with  a  uniform  twist  of  one  turn  in  25  calibers,  we 
have  p  =  7°  9'  45".  Employing  the  values  of  ju  and  /  already 
given,  (9)  reduces  to 

2R  =  0.063624  P (26) 

Working  Expressions. — If  the  equation  of  the  developed 
groove  is  (10),  we  have 

(        Mv2   > 

2  R  =  K  tan  B  -(  P  +  —, r  L      .      .      (27) 

L          2(u  +  a)} 

and 

TT  /  u •  +  a  \  \ 
tan  6  =   - 

25\w2+  a  / 

If  (16)  is  the  equation  of  the  developed  groove,  we  have 
and 


tan  9  = -I- 


l82  INTERIOR   BALLISTICS 

Pressure  on  the  Lands  of  the  lo-Inch  B.  L.  R.  —  The  equations 
for  velocity  and  pressure  for  this  gun  are  the  following: 

ir=  [6.20536]*!  {i  -  [8.59381  --  io]X0}    .      .     (29) 
and 

p  =  [4.720601*8  1  1  ~  [8-59381  "  io]*4}    .      .      (30) 

The  gun  and  firing  data  were  Vc=  7064  c.  i.,  um=  22.925  ft., 
co  =  250  Ibs.  of  brown  cocoa  powder,  w  =  575  Ibs.,  muzzle 
velocity  1975  f.  s.,  maximum  pressure  on  base  of  projectile, 
33300  Ibs.  per  in.2,  A  =  0.98,  and  z0=  3.461  ft.  It  will  be 
convenient  to  change  (30)  so  that  it  will  give  the  entire  pressure 
(P)  on  the  base  of  the  projectile;  and  to  avoid  large  numbers 
we  will  adopt  the  ton  as  the  unit  of  weight.  We  then  have 


- 
~  8960 

and  (30)  becomes 

P  =  [3.26544]  M1  -  [8-5938i  --  10]  X<\.      .     (30') 
Finally,  the  mass  of  the  projectile  expressed  in  tons  is 


We  have  now  all  the  formulas  and  data  necessary  for  comput- 
ing the  pressures  on  the  lands  of  the  lo-inch  B.  L.  R.,  by  means 
of  (26),  (27),  and  (28),  for  the  three  principal  systems  of  rifling 
adopted  in  our  service.  These  calculations  are  given  in  the 
table  on  page  183. 

The  last  three  columns  of  this  table  show  that  the  maximum 
pressure  on  the  lands  is  greater  for  uniform  twist  than  for  either 
form  of  increasing  twist;  but  the  difference  between  these  max- 
ima is  not  very  great.  Moreover,  the  maximum  pressure  for 
uniform  twist  occurs  at  the  trunnions  where  its  torsional  effect 
upon  the  gun  —  so  far  as  deranging  the  aim  is  concerned  —  is  a 


ON   THE   RIFLING   OF   CANNON 


Pressures  on  lands  required  to  produce  rotation  of  shot  in  the 
lo-inch  B.  L.  R.  for  different  systems  of  rifling.  Charge  250 
Ibs.  Projectile  575  Ibs.  Muzzle  velocity  1975  f.  s.  Maximum 
pressure  on  base  of  projectile  33300  Ibs.  per  square  inch. 


Travel 
of 

Velocity 
of 

Pressure 
on  Base  of 

PRESSL 

RE  ON  LANDS. 

TONS 

X 

Shot, 
feet 

Shot, 
f.  s. 

Shot, 
tons 

Uniform 
Twist 

Increasing 
Twist 
Eq.  (28) 

Increasing 

Twist, 
Eq.  (27) 

0.0 

0. 

0.0 

0.0 

0.0 

0.0 

0.0 

.1 

0.3461 

227.7 

841.1 

53-5 

I2.O 

27.0 

.2 

0.6922 

366.7 

1036.4 

65.9 

21.5 

36.9 

o-3 

1.0382 

478.1 

1122.4 

71.4 

29.0 

42.3 

•4 

1.3843 

572.4 

1158.3 

73-7 

35-3 

46.1 

•5 

1.7304 

654.7 

1167.4 

74-3 

40.9 

48.9 

0.6 

2.0765 

727.8 

1161.1 

73-9 

44.8 

5LI 

.7 

2.4226 

793-6 

1145.6 

72.9 

48.5 

52.9 

.8 

2.7686 

853-4 

1124.7 

71.6 

5L6 

54-3 

0.9 

3-II47 

908.2 

1100.7 

70.0 

54-3 

55-5 

I.O 

3.4608 

958.7 

1075.0 

68.4 

56.7 

56.5 

i.i 

3.8069 

1005.6 

1048.5 

66.7 

58.7 

57-3 

.2 

4.I530 

1049.3 

1021.9 

65-0 

60.5 

58.1 

•3 

44991 

1090.2 

995-5 

63-3 

62.1 

58.7 

•4 

4.8452 

1128.6 

969.7 

61.7 

63-5 

59-2 

•5 

5.I9I2 

1164.7 

944-5 

60.  i 

64.7 

59-7 

.6 

5-5372 

1198.9 

920.1 

58.5 

65.8 

60.  i 

•7 

5.8833 

1231.4 

896.4 

57-0 

66.7 

60.5 

.8 

6.2294 

1262.1 

873.6 

55-6 

67-5 

60.8 

•9 

6-5755 

1291.4 

851.7 

54-2 

68.2 

61.1 

2.0 

6.9216 

I3I94 

830.5 

52-8 

69.0 

61.3 

3-0 

10.3824 

15434 

6594 

42.0 

72.7 

62.4 

4.0 

13.8432 

1702.8 

541-4 

34-4 

73.5 

62.3 

5-o 

17.3040 

1824.9 

456.4 

29.0 

73-2 

61.6 

5.6586 

19.5833 

1891.6 

412.4 

26.2 

72.6 

61.0 

6.0000 

20.7648 

1922.8 

392.4 

25.0 

25.0 

25.0 

6.6242 

22.9250 

1975.0 

359-9 

22.9 

22.9 

22.9 

1 84  INTERIOR   BALLISTICS 

minimum;  while  the  position  of  the  maximum  pressure  upon  the 
lands  for  either  form  of  increasing  twist  is  well  down  the  chase. 
It  is  difficult  to  see  any  superiority  of  an  increasing  twist  over  a 
uniform  twist,  especially  in  view  of  the  fact  demonstrated  by 
Captain  Noble's  experiments,  that  the  energy  expended  in  giving 
rotation  to  the  projectile  with  rifling  having  an  increasing  twist 
is  nearly  twice  as  great  as  with  a  uniform  twist. 

Application  to  the  1 4-inch  Rifle. — This  gun  has  126  grooves 
and  the  same  number  of  lands,  in  this  respect  differing  from  the 
rule  followed  with  the  other  seacoast  guns.  The  values  of  n\9 
HZ,  61  and  02  are  the  same  as  those  found  for  the  lo-inch  rifle. 
The  rifling  begins  7.05  inches  from  the  base  of  the  projectile 
when  in  its  firing  seat  and  becomes  uniform  22.8  inches  from 
the  muzzle.  Therefore 

u2  =  4i3-85  ~  (7-°5  +  22-8)  =  384  inches. 
From  (12),  (13),  and  (14),  we  now  find 

a  =  128 

p  =  0.0037024 

b  =  5.36165 

Therefore  the  equation  of  the  developed  groove  is 

y  +  5-36165  =  0.0037024  (u  +  128)' 
From  (n)  and  (15),  we  have,  finally, 


tan  0  =  0.0055536^  u  +  128 


VU  +  I2S 

M  =  — — -  =  c 
2240  £ 

and 


in  which  P  is  the  entire  pressure  on  the  base  of  the  projectile  in 


ON   THE    RIFLING    OF   CANNON 


tons  while  p  is  the  pressure  in  pounds  per  square  inch  given  by 
the  equation  on  page  166,  for  a  charge  of  314  Ibs. 

Substituting  these  expressions  for  tan  0,/"  (x),  and  M  in  (25) 
and  reducing,  we  have  the  working  expression 

—I 

128 


2R  =  [7.44769-10]  V  I* -f  128  jP|4- [8.06145-10] 


in  which  2  R  is  the  normal  pressure  on  all  the  lands  in  tons  and 
u  the  travel  of  the  projectile  in  inches.  To  determine  the 
normal  pressure  on  each  land  2  R  must  be  divided  by  126. 

For  a  uniform  twist  2  R  is  given  by  (26). 

In  the  following  table  v  and  p  were  computed  by  the  formulas 
on  page  166  for  a  charge  of  314  Ibs.,  and  P  and  2  R  by  the 
formulas  given  above: 


M 

inches. 

V 

L  s. 

p 

tons. 

2   R 

Increasing  T 

2   R 

Uniform  T 

O.I 

4.72 

198.7 

1671.7 

54-i 

106.4 

0.2 
0-3 

9-45 
14.17 

325.2 
429.1 

2153-0 
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81.1 

137.0 
153-4 

0.4 
0-5 

0.6 

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28.34 

518-9 
598.5 
670.3 

2558.8 
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2686.3 

87-7 
92.2 

95-3 

162.8 
168.2 
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0.7 

0.8 
0.9 

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42.50 

735-8 
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101.8 

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141.68 
165.29 

1436.7 
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2049.3 
1927.6 

100.5 
99.1 

97-8 

139-3 
130-4 
122.7 

4.0 
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6.0 

188.91 
236.13 
283.36 

1756.5 
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66.0 

59.3 

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8.0 

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377-82 

413-85 

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787.0 
671.8 
598.8 

53-6 

48.8 

50.1 
42.7 

1  86  INTERIOR   BALLISTICS 

Influence  of  the  Rifling  for  a  Uniform  Twist.  —  For  a  uniform 
twist  we  have 


where  n  is  constant,  and  r  is  the  radius  of  the  projectile.     Differ- 
entiating with  respect  to  /  we  have 

do)       TT  dv        TT   dzjx 
~dt  "  "nr  dt~  nr  dtz 

Substituting  this  value  of  d  u/d  t  in  (3),  it  becomes 

*-^*J  =  2IZ(cos/J-/sm0)    .      .      .     (31) 
Eliminating  2  R  between  equations  (2)  and  (31),  gives 


P  M,. 

~-~dP\M          n      i-/tan/3j' 

If  the  bore  were  smooth  the  equation  of  translation  of  the 
projectile  would  be 


from  which  it  appears  that  the  effect  of  the  grooves  upon  the 
motion  of  the  projectile  for  a  constant  twist  is  equivalent  to 
increasing  the  mass  of  the  projectile  by  the  quantity 

TT_MM    /  +  tan  |8 
n        i  —  tan  0 

By  making  /  =  0.2,  n  =  25,  ^  =  0.5  and  /3  =  7°  9'  45",  the 
value  of  this  supplemental  term  is  found  to  be  0.021  M.  That 
is,  the  retarding  effect  of  a  constant  twist  of  one  turn  in  25 
calibers  is  equivalent  to  increasing  the  weight  of  the  projectile 
2  per  cent. 

*  Artillery  Circular,  N,  p.  201. 


TABLES 


PAGE 


I.  X  Functions l89 

II.  K  =  1-  (l-fc)i 2I1 

III.  Work  of  Fired  Gunpowder 2I2 


TABLES 


189 


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0  0  0  M  M 

INTERIOR   BALLISTICS 


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TABLES 


Q 

8  t^  to  co  O 
ON  ON  ON  ON 
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00  O   CO  O    ON 
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194 


INTERIOR   BALLISTICS 


d   M   04   01    01. 


I-H  o  oo  r—  to      ^  04  o  ON  r—     oiocoo4>-H       ONOO  r— o  ^h 

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04    04    04   O4    04          04O4O4O404          0404O4O4O4          O404O4O404 


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tO^J-        COCOO4I-HO         O   ONOO 


IO  IO  rf  CO         O4    O4    HH    O 


ON 


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ON  COO   ON  i-< 
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CO 


rj-  ON  COO   ON  t-i   04   04   O4   I-H  ONO   co  ON  Tt- 

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to  to  too  o      o  o  o  o  o      o  o  o  r—  r—      r—  r—  r—  r—  r—      r^^  r—  r— oo  oo 


TABLES 


195 


CQ 


q 

CON   O   ONOO 

HH     hH     HH     O     O 

o*  o5  o  o  o 

*"-*   O   ONOO   ts>* 
O   O   ON  ON  ON 

vO  10  rl-co  ON 
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CO  Tt-vO   t^  1^ 

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00  ON  ON>  O   O 
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O  O  O   O  O 

O4    04    O4    04    CO 
IO  IO  IO  IO  IO 

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O4    CO  IO  1^  1-1 
CO  CO  CO  CO  rf 
IO  IO  IO  IO  IO 

O  vO  ONOO  co 
iO  Ol   ONVO   co 
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I99 


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CO  CO  CO  co  CO 

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CO  CM    CM    CM    CM 
CO  CO  CO  CO  CO 

CO  CO  CO  fO  CO 

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00   CO  t^  CM  vO 

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co  t^  o  co  r^ 

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00  10  ONO  t- 
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HH  co  iO  t^  ON 
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ON  ON  ON  ON  ON 

66666 

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66666 

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tSSiS 

66666 

t^.0  HH  o  r>. 

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HH    O     ONt^iO 
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CM   CM    CM   CM   CM 

Si 

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HH  IO  HH  ON  ON 

VO  HH  ^  CM  00 
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00  00  OO    t^  t^ 

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CO    HH      ON   t^-    ^J" 

IO  IO  IO  IO  IO 

CM    HH    HH    CM   IO 
CM    O  00   IO  CO 
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4 

8 

& 

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ON  ON  ON  ON  ON 

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H 

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3 

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rj-  rt-  10  10  10 

HH   co  rf  CM   ON 
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HH  lOOO   ON  ON 
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Q 

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t^  rt-  ON  CM   rt- 
00    -^J-  ON  iO  O 
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ON  ON  ON  ON  O 

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o  100  100 

10  o  »o  o  10 

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too  100  »o 

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r^-  rl-  T|-  iO  iO 

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IO  IO  »O  IO  IO 

200 


INTERIOR   BALLISTICS 


q 

CS   ON  t^-vO  iO 
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HH     "3-00     HH     -^t- 

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X 
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HH  co  cs  ON  iO 

00  OO  00    t^vO 
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00   cOOO    CO  t^. 
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co  r^  HH  1000 

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O  IO  CN  00  rj- 
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^t-  Ti-  10  10  10 

00000 

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"8 


TABLES 


2OI 


w 

CQ 


Q 

ON  cor--  cooo 

hH  HH  O  O  ON 

COOO   CO  ONTt- 
ONOO  OO   t->-  t^ 
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ON  NO    CM   t^  CO 

NO  NO   NO     IO  »O 

CO  CO  co  CO  co 

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i-i  t^rt-CM  00 
CO  CM    CM    CM    I-H 
CO  CO  co  CO  co 

* 

0 

oc  t^o  r^o 
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10  1C  NO  NO  t>« 

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NO     CO  O   NO     l-< 

co  r^  I-H  Tt-oo 

g'gs2  2  2 

CM  CM  l^  ON  t^ 
t>.  CM  NO  O  rt- 
I-H  lOOO  CM  IO 

HH  HH  HH  CM  CM 

CM   COO   rf-NO 
00   HH   rt-NO  00 
00    CM   lOOO    I-H 
CM    CO  CO  CO  rf 

CM  cs  CM  CM'  CM 

CNJ    M    CM    CN    CNJ 

CM   CM    CM   CM   CM 

CM  CM  CM  CM  CM 

CM   CM   CM   CM   <S 

Q 

t^  IO  CM  ON  l^. 

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Tt-CN    ONVO   >0 

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NO     l>.  t^.  Is-  t^. 

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rhNO  iO  CO  ON 
rt-  CM  O  00  IO 
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O  O  O  O  O 

to  »ONO  to  N 
CO  O  t>.  rh  HH 

NO  00   ON  I-H   CO 

ON  ON  ON  O    O 

q 

00  rt  <->  00  rt 
Cs  CM  CM  i-i  HH 
fO  co  co  co  co 

1 

2^g>8 

cO  tO  cO  cO  cO 

t^  IO  CM    ON  t^ 

ON  ON  ONOO  OO 
CM    CM    CM    CM    CM 

rt-  CM  ON  t^  to 
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CM  CM  CM  CM  CM 

CM    O  00    IO  CO 

t^.  l^.VO   NO   NO 

CM    CM    CM    CM    CM 

« 

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9 

I-H  tO  ONOO  O 
rt  I-H  OC  O  IO 

CO  O  NO  co  O 
NO  NO  iO  iO  iO 

NO    ri-«0  ONVO 
CO  CM   1-1   O   O 

t^    Tf  I-H    00     IO 

rj-  rj-  rf  co  CO 

NO    ON  rl-  CM   CO 
O    O    >-H    CM    CO 
CM   ONNO   co  O 
CO  CM    CM    CM    CM 
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NO  CM  O  "H  rh 
rt-NO  00  O  CM 
t^  rj-  I-H  ONNO 

HH  I-H  hH  O  O 

rt-  t^  O    CO  t^ 
CO  O  00  iO  CM 
O   O   ON  ON  ON 
rt  rt-cO  COCO 

ON  ON  ON  ON  ON 

ON  ON  ON  ON  ON 

ON  ON  ON  ON  ON 

ON  ON  ON  ON  ON 

0-0,  ON  ON  ON 

r> 

CM  O  00  lOcO 
rt  rtcOCOCO 

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CO  CM    M    CM    CM 

HH     ON\O     10  CO 

CM  I-H  00  NO  iO 

3-808% 

i  * 

M 

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O  CM  O)  O  IO 

co  r^  I-H  1000 

00  ON  I-H  CM  CO 
HH  i-i  CM  fs  CNJ 

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•-i    TJ-  t^  O    CM 
IOVO    t^  ON  O 
CM    CM    CM    CM    CO 

O     I-H     O   NO     M 

IO  t^  O\  O    CM 
HH    CM    CO  IOVO 
CO  CO  CO  CO  CO 

rhNO  t>»  to  >-H 
CO  ri-  lONO  r^ 
t^.00  ON  O  I-H 
CO  CO  co  rl-  rt- 
t>-  t~>.  t"»  t^-  1^ 

NO   O   CM   CO  co 

t^.00  00  00  00 

CM   to  rt-iONO 
rj-  rt-  rf  rf-  rt- 

ON  ON  O  ON  ON 

ON  ON  ON  ON  ON 

ON  ON  ON  ON  ON 

ONONONO^ON 

ON  ON  ON  ON  ON 

Q 

ONTj-ONiOO 
04  (^  i_  _c  HH 
CO  CO  co  CO  CO 

IO  HH  00    CO  ON 
O   O   ON  ONOO 
CO  CO  CM    CM    CM 

IO  CM  00    rt-  HH 

oo  oo  r^  r>>  t>> 

CM    CM    CM    CM    CM 

tN.  rt-  HH  t^  rh 

NO  NO  NO  IO  IO 

CM  CM  CM  CM  CM 

I-H    ONNO    CM    ON 
iO  rt  rj-  rf  CO 
CM    CM    CM    CM    CM 

* 
g 

rj-  ro  t^vO  1-1 
00  HH  co  ID  f>- 
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M  \o  r>.  »ooo 

00  00  00  00   t^ 
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ON  ON  O    O    O 
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t>.  CM    rj-  CM  NO 

NO  iO  CO  HH  00 

HH     Tj-  l^-.  O     CM 
HH     I-H     P-H     CM     CM 

NO   NO   O   NO   NO 

l^.  rt-OO  ONNO 
»O  CM  00  rl-  O 
tOOO  O  CONO 
CM  CM  CO  CO  CO 

NO  NO  NO  NO  NO 

O   M   O  NO  00 
NO   i-i  NO   O   rt- 
OO   I-H   CONO  00 
CO  rt-  rt  rt  rt 

NO   Nfi   VO  NO   NO 

O  O  O  O  O 

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O  O  O  O  O 

00000 

O  O  O  O  O 

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t^  TJ-  fS  ON  t>. 

oo  oo  oo  r^  t^ 

10  CO  »-i    ONO 

r^  i>.  I^NO  NO 

»O  CO  >-i   ON  t~~- 

NO   NO   NO     IO  IO 

NO  rh  CM  I-H  ON 
IO  IO  IO  IO  rf 

t^O    IO  CO  CM 

c 

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5 

rj-  M  ior>.^o 

IO  rj-  CN  O  OO 
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00  00  00  00  00 

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NO   CO  M  oo   >O 
I-H    tO  iO\O  OO 

oo  oo  oo  oo  oo 

t^  CM    >ONO   IO 

I-H  00   rh  O  vO 
O   HH    CO  »O^O 
00  00  00  00  00 

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CM  00  CM  rt-  IO 

CM  f^  COOO  CO 
00  ON  I-H  CM  rt- 

00  00  ON  ON  O 
OO  OO  OO  OO  OO 

^j-  HH     J>»  CM     tO 

00   CO  t^  CM  NO 

10  r^oo  O  I-H 

ON  ON  ON  O    O 
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1-^00  ON  O  I-H 

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CM    CO  ^-"P^O 

d  o  o  o  o 

06  od  06  06  06 

00  00  00    ON  ON 

ON  ON  ON  ON  ON 

2O2 


INTERIOR   BALLISTICS 


^-  HH   00     *O   <*O  O     Is"*   ^^  M   00  ^^     CO   O 

NH  HH  O  O  O        O  ON  ON  O^oO       oo  oo  oo 
CO  CO  CO  CO  rO         CO  Ol   M    Cl   0*          (N    M    <N 


CO  t-H     ONt>. 

t-»  t>»O  vO  \ 
CM   W   CN   tN 


CN   O   l^.  tO  CO 

vO  vO   iO  iO  iO 
CN    CN    CN    CN    CN 


I 


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ON  NO   04  OO  cO  ON 

£3  w  ^hri.      6  covd  ONCN       tooo  6  covo       ONhnrft^ON  CNIOI^OCN 

^t-iOtOtO       VOvOvOvOt^«        l>»  t>-OO  COOO        OOONONONON  OOOhHhH 


OOONr>-CN  lOtOCNVOt^  lOhHTt-Tj-HH  VOON 

HHNCO^      ^"t^-cocN       hHOogvort-      HHOO 


CN    CN    CN    CN    CN 


CN    CN    CN    CN    CN 


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I 


cOCN         ONIOONCNCO        cOCNONiOON  CN<Or->.t->.iO  cOONcOt^O 

vO  CN         ^»  COOO   ^t"  ON        •<*•  ON  cOOO   CN  t^nnioONcO  1^-OTl-t^.HH 

O   HH   ro  -^-vo  t^  ON  O  hH  co  -^-vO  1^-00   O 

CNCNCNCNCN  CNCNCOcOcO  cOcOcOcOTt- 


CN  OM^  to  co 

yft   to  IO  iO  IO 
CN   CN    C-»   CN    CN 


I-H      O 


CN    CN    CN          CN    CN    CN    CN 


co  CN  O  t^vO  to  co  hH  O  ON 
COCOCOCNCN  CNCNCNCNhH 
CNCNCNCNCN  CNCNCNCNCN 


^ 

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co  co  co 


IO  rf  IO  t-»  0) 
(N  1^  C^  t^  CO 
t>» 


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t">«  rO  O  ^O   ^*O  O   t^1*  *-O  C^   O  00  ^O 

t">»  10  to  o  oo  *o  co  *^  cr\  i^*  T^  cs 

CO  rO        co  co  co  to  ro        to  ro  rO  rO  rO  co  to  ro  co  CO  ro  to  to  ro  CO 

ON  ON  ON  ON  ON  ON  ON  0s-  ON  ON 


ON  ON  ON  ON  ON 


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COCOCOCNCN  CNCNi-iMHH  hnnnOOO  OOONONON  ON  ONOO  00  00 
CNCNCNCNCN  CNCNCNCNCN  CNCNCNCNCN 


t^  rj-00  O   ON        to  ON  M   O 
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IO  IO  10  »O\ 


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CO  iO  t^  ON  hH          CO  iO  t^  ON  hH  CO  iO  t^  ON  >-< 

'  -00        OOOOOOOOON  ONONONONO 


00000    00000    00000    00000    00000 


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ON 
C4 


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ON 


ON  ON  ON 
60660 


fO  t^  ON  O  O 
ON  (N  iO  ON  0» 
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66666 


^t-  r^  O  o)  10 

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hH      M     M      (^     CN 

ON  ON  ON  ON  ON 
66666 


rj-  O 

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ON  CO 


lOr}-  CN 

t>.  ON  M  CO  »O  t>-C 

CN   CO  tOvO   l^  00   ON  HH    CN   CO 

CNCNCNCNCN  CNCNCOcOcO 

ON  ON  ON  ON  ON  ON  ON  ON  ON  ON 

66666  66666 


r^oo  ONO  M 


t^OO   ON  O  I-H 


t^oo  ON  O  M 


OOOhHHH  HHhHHHHHhH  HHHHHHCNCN  CNCNCNCNCN  CNCNCNfOrO 


TABLES 


203 


O4   OJ   04   04    04 


M  ONOO  O 
cotO 
CN)    M 


iO  ON  cONO  t-> 
10  Tf  rl-cOcO 
rh  Tf  rj-  rfrf 


§£'§•£ 

I-H  I-H  04  04  O4 
O4  04  O4  04  O4 
04  04  04  04  04 


N  CO  0<  O  O 
oo  M  vo  O  r-» 
t^  O  M  iO  ON 
C<  fO  fO  CO  ro 


tOOO  t^O  NO 

cooo  cooo  t-i 

•^-00   CO  t^  Cl 
<3- 


t^.  CO  iO  co  ON  M  oo  O  NO  00 
I-H  O  O  OO  O 
t^  i-i  IOOO  O4 


l^  ON  O   O 
O    -*•  ON  co 


M    04    04    04    04 


O    O    ONOO   t^ 
CO  CO  04    O4    Ol 


I-H  00  co  i-i  l^.  Tt-  i-HOOTJ-cO 
•<*•  COCOCOO4O4  C4i-i>-ii-H 
04  04O404O4O4  04  O4  04  Ol 


o"S   ON  ON" 

04     04     HH     HH 


VO  "H  10  ON  rO 
OO  i-i  ro  iO  O 
t>»  ON  O  M  ^f 
r^-  ri-  IO  IO  IO 


NO  00   HH    CO  iO 


t^OO  NOOcO  0100O4COO4 

ON  I-H    ro  lONO  »>»  t^OO  00  00 

t>.  O  01  ^t-O  oo  O  04  ThNO 

NO  i^*  t^»  t^  IN*  i>*oo  oo  oo  oo 


t^  »O  TJ-  co  HH 


ONOO  00  O 
O  O  O  I-H 


ON  iO  O  iO  i-i 
CO  ON  ONOO  00 
COCO  CO  CO  CO 


iN-i-it^coON       lOt-ir^coO 

r^t^NOOlO         IO  IO  Tl-  rf  rf 

cococococo        cococococo 


^ 

8 


O  00  NO  co  M 
CO  M  M  04  O4 
cOfO  fO  CD  to 


\D\O  00   O   O 
ON  t"**  iO  ro  C^ 

CO  coco  coco 
Q\  &\  QN  Qs  Q\ 


coc 
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coco 

C^1  ON  ON  O\  O^ 


looo  i^  o 


00    CO  Ol   »O  O4 

Tj-  ON  10 

r^  co  o  o  co 


IOCNOO-^-'-H       r^roo 

CS    CJ    CS    CNJ    OJ  CJ    CN]    CJ    CN |    Cl 

Os  ON  ON  ON  ON   ON  ON  0s-  O^  ON 


l^^O   O  00 

NO  NO   co  (N 


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W 

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ON  ON  ON  O  ON   ON  ON  ON  ON  ON   ON  ON  ON  ON  ON 


00  COOO  O4  00 
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cO  co  co  co  co 


r^-00  CO  ON  IO 
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O4    O4    04    04    O4 


00  NO  ON  f^  ON 
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ON  O 

ON 


HH     ON  04     HH  VO 

O   ON  ONOO        NO   Tj-  O4   ON\ 
CO  IOOO   HH          Tht^O   04 
IO  lONO  NO 


dodoo   ooooo   ooooo   ooooo   ooooo 


ON 


NO  00   ON  ONOO  NO   ^t  i-i   t^  t^  cONO  t^  ^-  ON 

^-  lONO   t^-OO  ONO>-ii-HO4  COCOCOCOO4 

rt-  lONO   t->.OO  ONi-iOlcOiO  I>.ON>-icOiO 

COCOCOCOCO  CO  -^-  •*  rj-  rj-  T^  rt- IO  iO  iO 

do  odd  ddodd  o'dddd 


HH     O     t^-   04      CO 

04    ^    ONOO  NO 
t^  ON  O    04    •* 

IO  »ONO  NO  NO 

ON  ON  ON  ON  ON 


co  O   IO 
TJ-  O4    ON 

00 


OO  00 

O   cO 

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ooooo   ooooo 


t>.00    ON  O    04 
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O   OJ 


q  o; 

iO  iO  lOO  O 


204 


INTERIOR   BALLISTICS 


w 

CQ 


Q 

n  00    ICOO    IO 

t^^O  sO   1C  i-O 
rO  fO  cO  cO  co 

>-   t^  CO  ON  1^- 
co  CO  CO  CO  CO 

cOOO  O  \£5   cO 

CO  04    0)    04    04 
coco  CO  CO  CO 

00   10  O  NO    co 

"I   •-.   «   O   O 

00    IO  >-i    t^  CO 

ON  ON  ONOO  OO 
Ot    O4    04    O4    04 

£ 

J 

to  t^  tOO  00 

ri-  I-H  oo  to  O 
\O   O   co  t^*  I-H 

co  co  fO  co  CO 

CO  ^t-  "-<   rt-  CO 

vO  «->  vO  O  TJ- 

TfOO    i-    UOOO 
O    CNJ    CO  CO  CO 
CO  COCO  CO  CO 

t^  O  00   ^t-  O 

t^.  HI   crjvo   ON 

W     1000      HH      Tj- 

Tt-  ^-  ^IO  IO 
CO  co  co  CO  CO 

co  I-H  vO  ^C   04 
I-H    CO  "sf  tOO 
00   M   •*  t^  O 
iO\O  \O  vO  l>> 
CO  co  co  co  co 

to  CO  X    O1  O 
NO  O    iO  r)-  CO 
COO    ON  O4   to 
t^  !>.  l^QO  00 
CO  CO  co  co  CO 

04    04    O4   04   04 

(N    CN    CN    M    CN 

04    04    04    O4    04 

04    O4    04    04    04 

O4    04    04    04    0) 

q 

co  04  o  o  «o 

ON  ON  ONOO  00 

CO  O   ONVQ   r}- 

oo  oo  t^*  t~^*  t^» 

Th  i-<  oo  »o  r^ 

!>.  I>-O     1>»NO 

CO  HH   QN  Q\  QN 
NO  ^O   to  iO  to 

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* 

ON04    ri-  Tt-O 

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00   O   O4    rJ-vO 

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CM  O  oo  ^o  •^r 

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t^  i-<    04    O   IO 

h-i    ON^O    CO  O 

t^oo  O  04  Th 

04   tOvO   uo  ^t- 
t^.  co  ON  to  HI 
IO  t^OO    O    O) 

COOO    CO  iO  rt- 

r^  01  co  coco 

rO  iOO  oc   ON 

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M    CS    O)    04    04 

O4    04    O4    O4    O4 

04    Ol    04    O4    04 

0)    0)    O)    O4    0) 

q 

O    04    ONO    04 
CO  CO  04    O4    O4 
<0  ro  rococo 
\ 

ONVD   CO  O  O 
_,   _,  _   _   o 
coco  coco  CO 

rt-  HI  oo  1000 

O  O   ON  ONOO 
CO  CO  04    O)    04 

IO    H-I    OO     l^-O 

ON  ONOO  OO  OO 
04    O4    04    04    O4 

O    COOO   iO  CO 

r-  t^-o  o  O 

04    04    04    O4    O4 

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04  vO   ^-  iO  ON 

M     (^    ^J-    |_    00 

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Ol   Ol   04    04    04 

1^00   04   ON  ON 

\O  Thco  w  O 
CO  O  t^  ^  i-" 
CO  CO  O)    04    O4 
04    04    04    04    04 

cO  ONOO   O  iO 
O   QN  ^s  O   O 
oo   "^  HH   o\^o 

04    O4    O4    04    04 

t^.  04    tn    CO\O 
HH    O4    CO  ^t  tO 
CO  O  t^  rj-  I-H 

%%222 

O    rj-  I-H    rOOO 
t^  ON  O4    IOOO 
oo  »o  co  O  r^ 

oo  oo  oo  oo  r^ 

ON  ON  ON  ON  ON 

ON  ON  ON  ON  ON 

ON  ON  ON  ON  ON 

ON  ON  ON  ON  ON 

ON  ON  ON  ON  ON 

q 

ON  ON  ON  ONOO 

t^.  l^.  IO  •<*•  O4 

oo  oo  oo  oo  oo 

04    O    ON  r^sO 
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r-  r^o  NO  o 

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ON  ON  ON  O    O 
t^  l^  t^OO  00 

O    04    04    HH  00 
I-H   C\t^  IO  O4 
04    O4    CO  -3-IO 

O  O  O  O   O 
oo  oo  oo  oo  oo 

Tf-ON^OO     I-H 
O    t^«  to  O4    O 

O4    CO  CO  04    O 

t-~  -4-  «  oo  to 

ON  C    I-H    HH    O4 
00  00  00  00  00 

ON  ON  ON  ON  ON 

ON  ON  ON  ON  ON 

ON  ON  ON  ON  ON 

ON  ON  ON  0,0, 

ON  ON  ON  ON  ON 

Q 

fS  OO  IO  O  OO 
vO  iO  ir;  10  T|- 
CS    CS    fN    CS    C4 

IO  04  00   IO  04 
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04    04    04    04    04 

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04    04    04    04    04 

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3^8:8; 

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| 

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O  t^  O4   iO  t^ 

04   t^.  COOO   CO 
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tlO  O4   l^«-  04 
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l^  hH     rf-  f^   ON 

HI  vO  O   Tt-00 
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ON  ON  O    O    O 
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t^  •*  O   t>-QO 

04  O   O   cOO 
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00000 

00000 

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00  00  OO   ON  ON 

ON  ON  ON  O    O 

O  O  O  "H  i-< 

|_!      |_l      HH      O)      O4 

TABLES 


205 


n  d  ON  t>-  lO 
\O  vO  lO  lO  10 
C4  W  M  CN  M 


ON  >-i   ON  >O  ON  (M   co  co  O  lOt^  ONOO  lO 

i-iot^-'O  oiOf^tt-'-i  r^.  co  ON  10  I-H 

OO  M   cO'O  ON  CM   ^}-  l^-  O  CM  to  l^-  O  co 

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TABLE   II 


211 


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0.65 

9.61108 

896 

0.918 

9.85348 

213 

0.984 

9.94127 

199 

0.66 

9.62004 

890 

0.920 

9-8556I 

215 

0.985 

9-94326 

205 

0.67 

9.62894 

886 

0.922 

9.85776 

217 

0.986 

9-94531 

211 

0.68 

9.63780 

882 

0.924 

9-85993 

218 

0.987 

9.94742 

219 

0.69 

9.64662 

879 

0.926 

9.86211 

221 

0.988 

9.94961 

227 

0.70 

9-6554I 

875 

0.928 

9-86432 

222 

0.989 

9.95188 

236 

0.71 

9.66416 

872 

0.930 

9.86654 

224 

0.990 

9-95424 

122 

0.72 

9.67288 

871 

0.932 

9.86878 

226 

0.9905 

9^5546 

125 

0-73 

9-68159 

869 

0-934 

9.87104 

229 

0.9910 

9.95671 

128 

0.74 

9.69028 

869 

0.936 

9.87333 

231 

0.9915 

9-95799 

132 

0-75 

9.69897 

869 

0.938 

9.87564 

234 

0.9920 

9-95931 

135 

0.76 

9.70766 

869 

0.940 

9.87798 

236   0.9925 

9.96066 

139 

0.77 

971635 

871 

0.942 

9.88034 

239   0.9930 

9-96205 

144 

0.78 

9.72506 

873 

0.944    9-88273 

242  |  0.9935 

9.96349 

150 

0.79 

9-73379 

877 

0.946    9-885I5 

245 

0.9940 

9.96499 

154 

0.80 

9.74256 

880 

0.948 

9.88760 

248 

0-9945 

9.96653 

162 

0.81 

9-75I36 

886 

0.950 

9.89008 

252 

0.9950 

9-96815 

I69 

0.82 

9.76022 

893 

0.952 

9.89260 

256 

0-9955 

9.96984 

I78 

0.83 

9.76915 

900 

0-954 

9.89516 

260 

0.9960 

9.97162 

I89 

0.84 

977815 

454 

0.956 

9.89776 

264 

0.9965 

9-97351 

203 

0.845 

9.78269 

456 

0.958 

9.90040 

269 

0.9970 

9-97554 

218 

0.850 

9.78725 

459 

0.960 

9.90309 

274 

3.9975 

9.97772 

241 

0-855 

9.79184 

462 

0.962 

9.90583 

28O 

0.9980 

9.98013 

272 

0.860 

9.79646 

465 

0.964 

9.90863 

285 

0.9985 

9.98285 

319 

0.865 

9.80111 

469 

0.966 

9.91148 

292 

0.9990 

9.98604 

414 

0.870 

9.80580 

473 

0.968 

9.91440 

300 

0.9995 

9.99018 

982 

0.875 

9.81053 

478 

0.970 

9.91740 

152 

I.OOOO 

o.ooooo 

.  .  . 

0.880 

9.81530   482 

0.971 

9.91892 

155 

0.885 

9.82012 

488 

0.972 

9.92047 

157 

0.890 

9.82500 

492 

0-973 

9.92204 

1  60 

0.895 

9.82992 

500 

0.974 

9.92364 

161 

0.900 

9.83492 

2OI 

0-975 

9-92525 

165 

0.902 

9-83693 

2O2 

0.976 

9.92690 

167 

0.904 

9.83895 

204 

0.977 

9.92857 

170 

0.906 

9.84099 

205 

0.978 

9.93027 

174 

14 


212 


INTERIOR   BALLISTICS 


TABLE  III. — Giving  the  total  work  that  dry  gunpowder  of 
the  W.  A.  standard  is  capable  of  performing  in  the  bore  of  a 
gun,  in  foot-tons  per  Ib.  of  powder  burned.1 


Number  of 
volumes  of 
expansion. 

Corresponding 
density  of 
products  of 
combustion. 

«1* 

Ms 

!<2J 

H  S.g 

Difference. 

Number  of 
volumes  of 
expansion. 

Corresponding 
density  of 
products  of 
combustion. 

Total  work 
per  Ib.  burned 
in  foot-tons. 

Difference. 

.OO 

I  .OOO 

.56 

.641 

T.A  !?OO 

8lQ 

.01 

.990 

.980 

.980 

•58 

•633 

OT*  •  O 

35-30^ 

•  uly 
.801 

.02 

.980 

1.936 

-956 

.60 

.625 

36.086 

•785 

•03 

.971 

2.870 

-934 

.62 

.617 

36.855 

•769 

.04 

.962 

3.782 

.912 

.64 

.610 

37.608 

•753 

•05 

•952 

4-674 

.892 

.66 

.602 

38.346 

-738 

.06 

•943 

5-547 

•873 

.68 

•595 

39.069 

•723 

.07 

•935 

6-399 

.852 

.70 

.588 

39-778 

.709 

.08 

.926 

7-234 

-835 

.72 

•  581 

40.474 

.696 

.09 

.917 

8.051 

.817 

•74 

•575 

41.1.56 

.682 

.IO 

•909 

8.852 

.810 

•76 

•  568 

41.827 

.67'! 

.11 

.901 

9-637 

-785 

•78 

.562 

42.486 

•659 

.12 

•893 

10.406 

.769 

.80 

•555 

43-133 

-647 

•13 

.885 

i  i  .  160 

•754 

.82 

•549 

43-769 

.636 

.14 

•877 

11.899 

•739 

.84 

•543 

44-394 

-625 

•15 

.870 

12.625 

.726 

.86 

•537 

45.009 

.615 

.16 

.862 

13-338 

•713 

.88 

•532 

45-6I4 

-605 

•17 

•855 

14.038 

.700 

.90 

•526 

46  .  209 

•595 

.18 

.847 

I4-725 

.687 

.92 

•521 

46.795 

-586 

19 

.840 

15.400 

-675 

•94 

•515 

47-372 

•577 

.20 

•833 

16.063 

.663 

.96 

.510 

47.940 

.568 

.21 

.826 

16.716 

-653 

.98 

•505 

48.499 

•559 

.22 

.820 

17-359 

•643 

2.00 

.500 

49.050 

•551 

•23 

.813 

17.992 

-633 

2.05 

.488 

50-383 

1  -333 

.24 

.806 

18.614 

.622 

2.10 

.476 

5I-673 

.290 

•s 

.800 
794 

19.226 
19.828 

.612 
.602 

2.15 
2.20 

•465 
•454 

52-922 
54-I32 

.249 

.,210 

•2l 

.787 

20.420 

-592 

2.25 

•444 

55-304 

.172 

.28 

.781 

21.001 

-581 

2.30 

•435 

56.439 

•135 

•29 

•775 

21.572 

•571 

2-35 

•425 

57-539 

.  100 

•30 

.769 

22.133 

-56i 

2.40 

•4T7 

58.605 

.066 

•32 

•758 

23-246 

1.113 

2-45 

.  .408 

59-639 

•034 

•34 

.746 

24-324 

i  .078 

2.50 

.400 

60  .  642 

i  .003 

•36 

•735 

25-37I 

1.047 

2-55 

•392 

61.616 

•974 

•38 

•725 

26.389 

1.018 

2.60 

•384 

62  .  563 

•947 

.40 
•42 

•7H 
.704 

27-380 
28.348 

.991 
.968 

2.65 
2.70 

•377 
•370 

63-486 
64-385 

•923 
•899 

•44 

.694 

29.291 

•943 

2-75 

•363 

65  .  262 

.877 

.46 

.685 

30.211 

.920 

2.80 

•357 

66.119 

-857 

.48 
•50 

.676 
.667 

31-109 
31  986 

.898 
.877 

2.85 
2.90 

•351 
•345 

66.955 
67.771 

.836 
.816 

•52 
•54 

.658 
.649 

32.843 
33-681 

•857 
-838 

2-95 
3-oo 

•339 
•333 

68.568 
69-347 

•797 
•779 

1  From  Noble  and  Abel's  "  Researches  on  Fired  Gunpowder 


TABLES 


2I3 


Number  of 
volumes  of 
expansion. 

Corresponding 
density  of 
products  of 
combustion. 

Total  work 
per  Ib.  burned 
in  foot-tons. 

Difference. 

Number  of 
volumes  of 
expansion. 

Corresponding 
density  of 
products  of 
combustion. 

"%  C  § 

iis 

321 

3- 

Difference. 

3-05 

.328 

70.109 

.762 

7.10 

.141 

105.125 

•539 

3.10 

.322 

70.854 

•745 

7.20 

-139 

105-655 

•530 

3-15 

•  317 

7L584 

•731 

7-30 

•137 

106.176 

•521 

3.20 

.312 

72.301 

.716 

7.40 

•135 

106.688 

•512 

3-25 

.308 

73.002 

.701 

7-50 

•133 

107.  192 

•504 

3-30 

.303 

73.690 

.688 

7.60 

•131 

107.688 

.496 

3-35 

.298 

74-365 

•675 

7.70 

.130 

108.177 

.489 

3-40 

.294 

75.027 

.662 

7-80 

.128 

108.659 

.482 

3-45 

.290 

75.677 

•650 

7.90 

.126 

109.133 

•474 

3-50 

.286 

76.315 

.638 

8.00 

•125 

.109.600 

•467 

3-55 

.282 

76.940 

.625  ! 

8.10 

.123 

110.060 

.460 

3.60 

.278 

77-553 

.613 

8.20 

.  122 

110.514 

•454 

3-65 

.274 

78.156 

.603 

8.30 

.  1  2O 

110.962 

.448 

3-70 

.270 

78.749 

•593 

8.40 

.119 

III.  404 

.442 

3-75 

.266 

79-332 

•583 

8.50 

.117 

III.  840 

•436 

3.80 

.263 

79-905 

•573 

8.60 

.116 

112.270 

•430 

3-85 

.260 

80  .  469 

•564 

8.70 

•US 

112.695 

•425 

3-90 

.256 

81.024 

•555 

8.80 

.114 

113.114 

.419 

3-95 

-253 

81.570 

•546 

8.90 

.  112 

113.528 

.414 

4.00 

.250 

82.107 

•537 

9.00 

.  Ill 

113-937 

.409 

4.10 

.244 

83.157 

1.050 

9.10 

.  no 

114  341 

.404 

4.20 

.238 

84.176 

i  .019 

9.20 

.109 

IH  739 

•398 

4-30 

.232 

85.166 

•990 

9-30 

.108 

II5-I33 

•394 

4.40 

.227 

86.128 

.962 

9.40 

.  I0o 

115-521 

.388 

4-50 

.222 

87.064 

.936 

9-50 

.105 

115-905 

•384 

4.60 

.217 

87.975 

.911 

9.60 

.104 

i  16.284 

•379 

4.70 

.213 

88.861 

.886  ! 

9.70 

.103 

116.659 

•375 

4.80 

.208 

89.724 

•  863 

9.80 

.  102 

117.029 

•370 

4.90 

.204 

90-565 

.841 

9.90 

.  101 

H7-395 

•  366 

5.00 

.200 

91-385 

.820 

10 

.  100 

H7-757 

•  362 

5.10 

.196 

92.186 

.801 

II 

.091 

121.165 

3.408 

5.20 

.192 

92.968 

.782 

12 

.083 

124.239 

3-074 

5-30 

.188 

93-732 

-764 

13 

.077 

127.036 

2.797 

5-40 

.185 

94-479 

-747 

H 

.071 

129.602 

2.566 

5-50 

.182 

95-2io 

•731 

15 

.066 

131.970 

2.368 

5.60 

.178 

95-925 

-715 

16 

.062 

134-168 

2.198 

5-70 

•175 

96.625 

.700 

17 

•059 

136.218 

2.050 

5.80 

.172 

97.310 

-685 

i   l8 

.055 

138.138 

.920 

5-90 

.169 

97.981 

.671 

!  19 

.052 

139-944 

.806 

6.00 

.165 

98-638 

-657 

20 

.050 

141.647 

•703 

6.10 

.154 

99.282 

.644 

21 

.047 

143-258 

.611 

6.20 

.161 

99.9I5 

•633 

22 

•045 

144.788 

•530 

6.30 

•159 

100.536 

.621 

23 

•043 

146.242 

•454 

6.p 

.I55 

!     101.145 

.609 

24 

.042 

147.629 

-387 

6.50 

-154 

101.744 

•599 

25 

.040 

148.953 

!  1-324 

6.60 

•151 

102.333 

•589 

30 

033 

154.800 

1  5-847 

6.70 

.149 

102.912 

•579 

35 

.028 

159.667 

4.867 

6.80 

.147 

103.480 

.=68 

40 

.025 

163.828 

4.  161 

6.90 

•145 

104.038 

•558 

45 

.022 

167.456 

3.628 

7.00 

•143 

104.586 

•548 

50 

.020 

170.671 

3-215 

WORKS    CONSULTED 


HUTTON:  "Mathematical  Tracts,"  London,  1812. 

RUMFORD:  "Experiments  to  Determine  the  Force  of  Fired  Gunpowder," 
London,  1797. 

RODMAN:  "Experiments  on  Metal  and  Cannon  and  Qualities  of  Cannon 
Powder,"  Boston,  1861. 

NOBLE  AND  ABEL:  "Researches  on  Explosives,"  London,  1874,  1879. 

NOBLE:  "On  the  Energy  Absorbed  by  Friction  in  the  Bores  of  Rifled  Guns." 
Reprinted  as  "Ordnance  Construction  Note,"  No.  60.  "On  Methods 
that  have  been  Adopted  for  Measuring  Pressures  in  the  Bores  of  Guns," 
London,  1894.  "Researches  on  Explosives."  Preliminary  Note, 
London,  1894. 

OFFICIAL:  "English  Text-book  of  Gunnery."     Editions  of  1897  and  1902. 

SARRAU:  "Recherches  sur  les  effets  de  la  poudre  dans  les  Armes,"  and  "For- 
mules  pratiques  des  vitesses  et  des  pressions  dans  les  Armes."  A 
translation  of  these  memoirs  into  English  by  Lieutenants  Meigs  and 
Ingersoll  is  given  in  Vol.  X  of  the  Proceedings  U.  S.  Naval  Institute. 
"Recherches  theoriques  sur  le  chargement  des  bouches  a  feu."  Transla- 
tion by  Lieutenant  Howard,  O.  D.,  in  "Ordnance  Construction  Note," 
No.  42.  "War  Powders  and  Interior  Ballistics."  A  translation  by 
Lieutenant  Charles  B.  Wheeler,  O.  D.,  as  "Notes  on  the  Construction 
of  Ordnance,"  No.  67.  Washington,  1895. 

Gossox  AND  LIOUVILLE:  "The  Ballistic  Effects  of  Smokeless  Powders  in 
Guns."  Translated  by  Major  Charles  B.  Wheeler,  O.  D.  "Notes  on 
the  Construction  of  Ordnance,"  No.  88.  Washington,  1906. 

DUNN:  "Interior  Ballistics."  Part  I.  "Notes  on  the  Construction  of 
Ordnance,"  No.  89.  Washington,  1906. 

SOUICH:  "Poudres  de  Guerre.     Balistique  Interieur,"  Paris,   1882. 

BAILLS:  "Traite  de  Balistique  Rationnelle,"  Paris,  1883. 

LONGRIDGE:  "Internal  Ballistics,"  London,  1889. 

MEIGS  AND  INGERSOLL:  "Interior  Ballistics,"  Annapolis,  1887. 

PASHKIEVITSCH:  "Interior  Ballistics."  Translated  from  the  Russian  by 
Captain  Tasker  H.  Bliss,  A.  D.  C.  Washington,  1892. 

BERGMAN:  "Larobok  i  Artilleriteknik."  Del  I.  Krutlara.  Stockholm, 
1908.  A  part  of  this  work  was  translated  for  the  author  by  Colonel 
Lundeen,  Coast  Artillery  Corps,  U.  S.  Army. 

GLENNON:  "Velocities  and  Pressures  in  Guns,"  Annapolis,  1889. 

CROZIER:  "On  the  Rifling  of  Guns."     Ordnance  Construction.     Note  No.  49. 

A.  W.:  "Des  Armes  de  guerre  Modernes  et  de  leurs  Munitions."  Revue 
Militaire  Beige,  Vol.  II,  1888. 

McCuLLOCH:  "Mechanical  Theory  of  Heat,"  New  York,  1876. 

PEABODY:  "Thermodynamics  of  the  Steam  Engine,"  New  York,  1889. 

RONTGEN:  "The  Principles  of  Thermodynamics."  Translated  from  the 
German  by  Professor  A.  Jay  Du  Boise.  New  York,  1889. 

LISSAK:  "Ordnance  and  Gunnery,"  New  York,  1907. 

Encyclopaedia  Britannica,  eleventh  edition,  1911. 

215 


INDEX 

ABSOLUTE  temperature,  definition  of,  17;   of  fired  gunpowder,  40,  47. 

Adiabatic  expansion,  definition  of,  26. 

Air  space,  initial,  definition  of,  76;    expressions  for  reduced  length  of,  76,  77, 

92,  94- 

Angular  acceleration,  174. 

Applications  of  velocity  and  pressure  formulas:  To  magazine  rifle,  130  to  134; 
to  Hotchkiss  57  mm.  rapid-firing  gun,  125  to  130;  to  6-inch  English  gun, 
115  to  125  and  140  to  147;  to  6-inch  Brown  wire  gun,  150  to  162;  to  8-inch 
rifle,  102  to  no;  to  lo-inch  rifle,  179;  to  i^-inch  rifle,  162  to  169  and  184; 
to  hypothetical  7 -inch  gun,  in,  136. 

Artillery  circulars  M  and  N,  references  to,  88,  135,  186. 

Atmospheric  pressure,  value  of,  21. 

Axite,  form-characteristics  of,  61. 

BALLISTIC  pendulum,  2,  3. 
Ballistite,  61,  140. 

Binomial  formulas  for  velocity  and  pressure,  112. 
Bliss,  Captain  Tasker  H.,  53. 
Board  of  Ordnance,  reference  to,  151. 
Boyle,  Robert,  15. 

B  N  powders,  form-characteristics  of,  61 ;  computation  of  by  velocity  for- 
mula, 129. 

CAVALLI,  reference  to,  8. 

Centervall,  law  of  combustion,  79. 

Chamber,  reduced  length  of,  31 ;   alignment  of  grains  in,  71;  effect  cf  varying 

volume  of,  in,  112,  166. 
Characteristic  equation  of  gaseous  state,  17. 
Characteristics  of  a  powder,  94. 
Charge  of  powder,  behavior  of  when  ignited  in  a  gun,  12,  13;  in  a  close  vessel, 

12,  35;   initial  surface  of,  73,  77. 

217 


21 8  INDEX 

Chase,  excessive  pressure  in,  55,  158. 

Chevreul,  reference  to,  8. 

Chronograph,  Noble's,  116;    Boulenge'-Breger,  126. 

Coefficient  of  expansion  of  a  perfect  gas,  16. 

Combustion  of  a  grain  of  powder,  11;  under  constant  pressure,  55,  79;  under 
variable  pressure,  79,  80. 

Composition:  of  gunpowder,  i;  of  cordite,  n,  117,  124;  magazine  rifle  powder, 
131;  ballistite,  140. 

Constants,  physical,  adopted,  92,  94. 

Cordite,  composition  of,  n,  117,  124;   form-characteristics  of,  63. 

Cube,  form-characteristics  of,  61. 

Cylindrical  grains:  solid,  63;  with  axial  perforation,  65;  with  seven  perfora- 
tions (m.p.  grains),  66  to  72. 

D'ARCY'S  method  of  experimenting,  4. 

Density:  of  powder,  1 1 ;  of  a  gas,  21 ;  of  loading,  37,  75,  77. 

Dulong  and  Petit,  law  of,  21,  23. 

ELSWICK  works,  mention  of,  115. 

Encyclopaedia  Britannica,  eleventh  edition,  reference  to,  115. 

Energies  neglected  in  deducing  equation  for  velocity,  121. 

Energy  of  translation  of  projectile,  32,  51,  52,  53,  80,  144. 

English  Text-Book  of  Gunnery,  reference  to,  115. 

Euler,  mention  of,  88;  equations  of,  174. 

Examples:    of  expansion  of  gases,  28;    of  the  formulas  of  Chapter  III,  77; 

relating  to  8-inch  rifle,  109;  to  6-inch  gun,  122,  124,  144,  155;   to  14-inch 

rifle,  1 66. 
Expansion,  work  of:  isothermal,  25;  adiabatic,  26;  in  the  bore  of  a  gun,  30,  47. 

FACTOR  of  effect,  49,  52,  54. 

Force  of  the  powder,  33,  36. 

Formulas:  Characteristic  equation  of  gaseous  state,  17.  For  specific  heat  under 
constant  volume,  22.  For  work:  of  an  isothermal  expansion,  25;  adia- 
batic expansion,  26,  27,  32;  of  gases  of  fired  gunpowder,  49.  For  tempera- 
ture: of  an  adiabatic  expansion  of  a  perfect  gas,  26,  27;  of  gases  of  fired 
gunpowder,  47.  For  pressure:  isothermal,  15,  17;  adiabatic,  27;  gases 
of  fired  gunpowder  in  close  vessels,  6,  36,  39;  in  guns,  45.  For  pressure 
in  guns  with  smokeless  powders:  While  powder  is  burning,  85,  86,  101,  106, 


INDEX  219 

112,  140,  152;  after  powder  is  all  burned,  86,  102.  Maximum  pressure, 
91,  101,  1 06.  Initial  pressure  when  powder  is  all  burned  before  projectile 
moves,  86,  87,  93,  94,  99.  For  velocity  of  projectile  in  guns  with  smokeless 
powders:  while  powder  is  burning,  83,  84,  89,  100,  101,  103,  112;  after 
powder  is  all  burned,  84,  85,  102.  For  limiting  velocity,  85,  93,  94,  98, 
101,  127,  141,  146,  150,  164.  For  computing  f,  32,  36,  84,  93,  94,  96,  97, 
106,  164.  For  vc,  91,  92,  93,  94.  For  k  and  k' ',  58,  59,  61,  62,  63,  65, 
68,  69,  72,  89,  90,  109,  136,  141,  148.  For  M,  M' ,  N  and  N',  84,  85,  93, 
94,  95,  97,  101,  102,  106,  112,  113,  114,  115,  118,  136,  139,  150,  155,  161, 
165.  For  y,  39,  83,  93,  132,  146.  For  a,  A,  fJ.,  58,  59,  60,  61,  62,  63,  64, 
65,  67,  68,  69,  84,  129,  134.  For  P',  86,  87,  93,  94,  98.  For  the  X 
functions,  87,  88,  89,  For  X0  °r  ^o'»  84,  93,  94,  119,  136,  149,  150,  152, 
164.  Working  formulas,  77,  93,  94,  95,  97.  For  inclination  of  groove, 
171,  172,  177.  For  pressure  on  lands,  175,  176,  180,  181,  185.  For 
semi-cubical  parabola,  176.  Common  parabola,  178. 
Frankford  arsenal,  mentioned,  130. 

GAS,  perfect,  17. 

Gay-Lussac,  law  of,  16;   mentioned,  8. 

Gossot,  Colonel  F.,  law  of  combustion,  79;   igniter,  151. 

Graham,  mentioned,  8. 

Grains  of  powder,  combustion  of  under  constant  pressure,  55 ;  vanishing-sur- 
face, 56;  volume  burned,  57;  form-characteristics,  58;  their  relation  to 
each  other,  58,  59. 

Granulation,  151,  163. 

Groove,  developed,  171;  width  of,  179. 

Gun-cotton,  10,  n. 

Gunpowder,  i,  2. 

HAMILTON,  Captain  Alston,  length  of  m.p.  grains,  71. 

Heat:   mechanical  equivalent  of,  18;   specific  heats,  18,  19,  21,  22. 

Hugoniot,  law  of  combustion,  79. 

Hutton,  Dr.  Charles,  experiments  with  gunpowder,  3,  4. 

INFLAMMATION  of  a  grain  and  charge  of  powder,  n,  12,  13. 
Isothermal  expansion,  25. 

JOURNAL  U.  S.  Artillery,  references  to,  71,  79,  125,  148. 


22O  INDEX 

LANDS,  width  of,  179. 

Lenk,  General  von,  experiments  with  gun-cotton,  10. 

Liouville,  R.,  law  of  combustion,  79. 

Lissak,  Colonel  O.  M.,  ordnance  and  gunnery,  29;   construction  of  velocity 

and  pressure  curves,  144. 
Longridge,  Atkinson,  loss  of  energy  in  gun,  53. 

MAGAZINE  rifle,  description  of,  130. 
Marriotte,  law  of,  15. 
Maximum  pressure  in  a  gun,  90,  91 
Maximum  value  of  X3,  90,  101. 
Mayevski,  mention  of,  8. 
Monomial  formulas,  100. 

Muzzle  velocities  and  pressures,  computed,  107,  120,  123,  124,  129,  130,  133, 
134,  143,  161,  169. 

NATURE,  reference  to,  115. 

Neumann,  mentioned,  8. 

Nobel,  N.  K.  powder,  law  of  combustion  for,  79. 

Noble  and  Abel,  experiments  with  fired   gunpowder  in    close  vessels,  and 

deductions  therefrom,  33  to  54. 
Noble,  Sir  Andrew,  experiments  with  6-inch  gun,  115;   coefficient  of  friction, 

1 80. 
Notation,  15,  17,  19,  23,  24,  31,  51,  56,  58,  60,  67,  72,  74,  75,  76,  79,  80,  81, 

82,  83,  84,  85,  86,  91,  92,  108,  149,  171,  172,  174,  176. 
Notes  on  the  construction  of  ordnance,  reference  to,  102. 

ORDNANCE  Department,  reference  to,  130,  162,  166. 
Otto,  mentioned,  8. 

PARALLELOPIPEDON,  form-characteristics  of,  60. 

Pashkievitsch,  Colonel,  lost  work  in  a  gun,  53. 

Piobert,  mentioned,  8. 

Point  of  inflection  of  X3,  101. 

Powder  grains.     See  Grains  of  powder. 

Powder,  smokeless.     See  Composition. 

Pressure:  of  fired  gunpowder  in  close  vessels,  6,  7,  9,  35,  37;  in  guns,  41. 

RADIUS  of  gyration  of  projectile,  174,  180,  186. 
Retarding  effect  of  uniform  twist,  186. 


INDEX  221 

Rifling  of  cannon,  advantages  of,  170. 

Robins,  Benjamin,  experiments  with  fired  gunpowder,  2. 

Rodman,  Captain  T.  J.,  experiments  with  fired  gunpowder,  8;  perforated 
grains,  9;  cutter  gauge,  9. 

Rumford,  Count,  experiments  with  fired  gunpowder,  4;  comparison  of  re- 
sults with  those  of  Noble  and  Abel,  6. 

SAINTE-ROBERT,  Count  de,  law  of  combustion,  79. 

Sandy  Hook,  mention  of,  151,  160. 

Sarrau,  ET.,  law  of  combustion,  80;  monomial  formula  for  pressure  in  a  gun,  95. 

Schonbein  of  Basel,  discoverer  of  gun-cotton,  10. 

Sebert,  law  of  combustion,  79. 

Spherical  grains,  form-characteristics  of,  59. 

Springfield  Armory,  mentioned,  130,  131. 

TABLES,  in  text:  of  specific  heats  of  certain  gases,  22;  of  pressures  in  guns  of 
fired  gunpowder,  46;  of  velocities  and  pressures  in  guns,  104.  107,  129, 
130,  133,  134,  143,  161,  165,  169,  183,  185;  of  pressure  on  lands,  183,  185. 

Temperature  of  fired  gunpowder,  45. 

Trinomial  formulas,  138. 

Twist,  uniform,  171;    increasing,  171. 

VIEILLE,  law  of  combustion,  79. 

WEAVER,  General  E.  M..  Notes  on  explosives,  referred  to,  n. 
Work  of  fired  gunpowder,  47. 
Working  formulas,  77,  92,  181,  185. 


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